Wavelet analysis of conservative cascades - Mathematics
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WAVELET ANALYSIS OF CONSERVATIVE CASCADES SIDNEY RESNICK, GENNADY SAMORODNITSKY, ANNA GILBERT, AND WALTER WILLINGER Abstract. A conservative cascade is an iterative process that fragments a given set into smaller and smaller pieces according to a rule which preserves the total mass of the initial set at each stage of the construction almost surely and not just in expectation. Motivated by the importance of conservative cascades in analyzing multifractal behavior of measured Internet traffic traces, we consider wavelet based statistical techniques for inference about the cascade generator , the random mechanism determining the re-distribution of the set’s mass at each iteration. We provide two estimators of the structure function, one asymptotically biased and one not, prove consistency and asymptotic normality in a range of values of the argument of the structure function less than a critical value. Simulation experiments illustrate the asymptotic properties of these estimators for values of the argument both below and above the critical value. Beyond the critical value, the estimators are shown to not be asymptotically consistent.
1. Introduction A multiplicative cascade is an iterative process that fragments a given set into smaller and smaller pieces according to some geometric rule and, at the same time, distributes the total mass of the given set according to another rule. The limiting object generated by such a procedure generally gives rise to a singular measure or multifractal – a mathematical construct that is able to capture the highly irregular and intermittent behavior associated with many naturally occurring phenomena, e.g., fully developed turbulence (see [10, 12, 4, 15] and references therein); spatial rainfall [6]; the movements of stock prices [14]; and Internet traffic dynamics [19, 3]. The generator of a cascade determines the re-distribution of the set’s total mass at every iteration; it can be deterministic or random. Cascade processes with the property that the generator preserves the total mass of the initial set at each stage of the construction almost surely and not just in expectation are called conservative cascades and are the main focus of this paper. Originally introduced by Mandelbrot [13] (also in the turbulence context), conservative cascades have recently been considered in [3] for use in describing the observed multifractal behavior of measured Internet traffic traces. In particular, Feldmann et al. [3] build on empirical evidence that measured Internet traffic is consistent with multifractal behavior by illustrating that “... data networks appear to act as conservative cascades!” They demonstrate that • multiplicative and measure-preserving structure becomes most apparent when analyzing measured Internet traces at a particular layer within the well-defined protocol hierarchy The visits of Sidney Resnick to AT&T Labs–Research were supported by AT&T Labs–Research and a National Science Foundation Grant from the Cooperative Research Program in the Mathematical Sciences. S. Resnick and G Samorodnitsky were also partially supported by NSF grant DMS-97-04982 and NSA grant MDA904-98-1-0041 at Cornell University. c
0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page
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S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
of today’s Internet Protocol (IP)-based networks, namely the Transport Control Protocol (TCP) layer and at the level of individual TCP connections, • this structure is recovered at the aggregate level (i.e., when considering the superposition process consisting of all IP packets generated by all active TCP connections) and causes aggregate Internet traffic to exhibit multifractal behavior. Well short of providing a physical explanation to the all-important networking question of “Why do packets within individual TCP connections conform to a conservative cascade?” the work in [3] is empirical in nature and relies on a number of heuristics for inferring multifractal behavior from traces of measured Internet traffic. However, to provide a more solid statistical basis for empirical studies of multifractal phenomena, progress in the area of statistical inference for multiplicatively generated multifractals is crucial. In this paper we contribute to the effort of providing rigorous techniques for multifractal analysis by investigating wavelet-based estimators for conservative cascades (i.e., for the class of multifractal processes generated by conservative cascades) and studying their large sample properties. In essence, the inference problem for conservative cascades consists of deducing from a single realization of the cascade process the distribution of the cascade generator that was presumably used to generate the sample or signal at hand. Intuitively, the generator’s distribution can be inferred from the degree of variability and intermittency exhibited locally in time by the signal under consideration. It can be expressed mathematically in terms of the local H¨older exponents which in turn characterize the singularity behavior of a signal locally in time. Moreover, since the local H¨older exponent at a point in time t0 describes the local scaling behavior of the signal as we look at smaller and smaller neighborhoods around t0 , a wavelet-based analysis that fully exploits the time- and scale-localization ability of wavelets proves convenient and is tailormade for our purpose. On the one hand, we exploit here the fact that the singularity behavior of a process can (under certain assumptions) be fully recovered by studying the singularity behavior in the wavelet domain; i.e., by investigating the (possibly) time-dependent scaling properties of the wavelet coefficients associated with the underlying process in the fine-time scale limit. On the other hand, using Haar wavelets, the discrete wavelet transform of a conservative cascade can be explicitly expressed in terms of the cascade’s generator (see for example [5]) and hence provides a promising setting for relating the local scaling behavior of the sample to the distribution of the underlying conservative cascade generator. In particular, we relate the distribution of the generator to an invariant of the cascade, namely the structure function or modified cumulant generating function (also known as Mandelbrot-Kahane-Peyri´ere (MKP) function [7]) and study the statistical properties (i.e., asymptotic consistency, asymptotic normality, confidence intervals) of two wavelet-based estimators of this function. Although the results in this paper have been largely motivated by our empirical investigations into the multifractal nature of measured Internet traffic [3, 5], we have clearly benefited from the recent random cascade work of Ossiander and Waymire [17]. Compared to the conservative cascades considered in this paper, random cascades are multiplicative processes with generators that preserve the total mass of the initial set only in expectation and not almost surely. This apparently minor difference ensures independence within and across the different stages of a random cascade construction but gives rise to subtle dependencies inherent in conservative cascades. Ossiander and Waymire [17] study the large sample asymptotics of estimators that are defined in the time-domain rather than in the wavelet-domain and allow for a rigorous statistical analysis of the scaling behavior exhibited by random cascades (for related work, see also [20]). While
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
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the large-sample properties of the time domain-based estimators considered in [17] and of the wavelet-based estimators studied in this paper are very similar, their potential advantages, disadvantages and pitfalls when implementing and using them in practice require further studies. However, in combination, these different estimators provide a set of statistically rigorous techniques for multifractal analysis of highly irregular and intermittent data that are assumed to be generated by certain types of multiplicative processes or cascades. The rest of the paper is organized as follows. Sections 2–4 contain the basic facts about conservative cascades, their wavelet transforms, and some related quantities that are needed later in the paper. Section 5 discusses the critical constants and Section 6 is concerned with certain martingales and leads into Section 7 where subcritical asymptotics (that is, asymptotics for values of the argument below the critical value) and strong consistency of our two wavelet-based estimators is established. Asymptotic normality of the estimators is explained and illustrated with some simulated data in Section 8. We conclude in Section 9 with some supercritical asymptotics when the value of the arguement exceeds the critical value. The values of the estimators at large values of the argument of the structure function are uninformative and misleading, thus providing some practical guidance for properly interpreting the plots associated with the estimation procedure. 2. The Conservative Cascade. We now summarize the basic facts about the conservative cascade. Consider the binary tree. Nodes of the tree at depth l will be indicated by (j1 , . . . , jl ) ∈ {0, 1}l . Alternatively we consider successive subdivisions of the unit interval [0, 1]. After subdividing l times we have equal subintervals of length 2−l indicated by (2.1)
I(j1 , . . . , jl ) =
l l hX 1 jk X jk , + , 2k 2k 2l k=1
k=1
(j1 , . . . , jl ) ∈ {0, 1}l .
An infinite path through the tree is denoted by
j = (j1 , j2 , . . . ) ∈ {0, 1}∞
and the first l entries of j are denoted by
j|l = (j1 , . . . , jl ). We will sometimes write when convenient j|l, jl+1 = (j1 , . . . , jl , jl+1 ). The conservative cascade is a random measure on the Borel subsets of [0, 1] which may be constructed in the following manner. Suppose we are given a random variable W , called the d cascade generator , which has range [0, 1] and which is symmetric about 1/2 so that W = 1 − W. The symmetry implies that E(W ) = 1/2. We assume the random variable is not almost surely equal to 1/2. There is a family of identically distributed random variables {W (j|l), j ∈ {0, 1}∞ , l ≥ 1}
each of which is identically distributed as W . These random variables satisfy the conservative property (2.2)
W (j|l, 1) = 1 − W (j|l, 0).
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S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
Random variables associated with different depths of the tree are independent and random variables of the same depth which have different antecedents in the tree are likewise independent. Dependence of random variables having the same depth is expressed by (2.2). The conservative cascade is the random measure µ∞ defined by (2.3)
µ∞ (I(j|l)) =
l Y
W (j|i).
i=1
Note the conservative property entails that (2.4)
µ∞ (I(j|l, 0)) + µ∞ (I(j|l, 1)) = µ∞ (I(j|l)),
so that the weight of two offspring equals the weight of the parent. This implies X (2.5) µ∞ I(j|l) = 1. j|l
3. Wavelet Coefficients. We compute the wavelet transform Z 1 (3.1) ψ−l,n (x)µ∞ (dx), d−l,n =
n = 0, . . . , 2l − 1; l ≥ 1,
0
using the Haar wavelets
( 2l/2 , ψ−l,n (x) := −2l/2 ,
(3.2)
if if
2n ≤ x < 2n+1 , 2l+1 2l+1 2n+2 2n+1 ≤ x < 2l+1 . 2l+1
We have by examining where the Haar wavelet is constant that 2n + 1 2n + 2 2n 2n + 1 l/2 d−l,n = 2 µ∞ [ l+1 , l+1 ) − µ∞ [ l+1 , l+1 ) . 2 2 2 2 P Now suppose that lk=1 jk /2k = n/2l . Then we have from the definition (2.3) d−l,n =2l/2
l hY i=1
=2l/2
l Y i=1
W (j|i)W (j|l, 0) −
l Y i=1
i W (j|i)W (j|l, 1)
h i W (j|i) W (j|l, 0) − W (j|l, 1)
and using the conservative property (2.2), this is l/2
(3.3)
d−l,n =2
l Y i=1
for n = (3.4)
0, 1, . . . , 2l
h i W (j|i) 2W j|l, 0) − 1 ,
− 1. Sometimes where convenient, we will also write l h i Y W (j|i) 2W j|l, 0) − 1 . d−l,n = d(−l, j|l) = 2l/2 i=1
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
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4. Notation Glossary. Before continuing the analysis, we collect some notation in one place for easy reference. We seek to estimate the distribution of the cascade generator W and this will be accomplished if we estimate c(q) := 2E(W q ),
(4.1)
q > 0,
or equivalently we could estimate the structure function (4.2)
τ (q) = 1 + log2 E(W q ) = log2 c(q).
The structure function will be estimated using estimators constructed from the process (4.3)
Z(q, l) =
l XY j|l i=1
W (j|i)q |2W (j|l, 0) − 1|q
and note from (3.3) that Z(q, l) =
(4.4)
1 2ql/2
l −1 2X
n=0
|d−l,n |q .
Our analysis rests on the process M (q, l), which we will show to be a martingale and which is defined as l 1 XY M (q, l) = (4.5) W (j|i)q , q > 0, l ≥ 1, c(q)l j|l i=1
and note the normalization makes E(M (q, l)) = 1. There are further constant functions needed: (4.6)
b(q) =E|2W − 1|q ,
(4.7)
a(q) =
(4.8)
ar (q) =
c(2q) E(W 2q ) = 2 , c2 (q) 2 E(W q )
21−r E(W rq ) c(rq) r . = cr (q) E(W q )
Note that a(q) = a2 (q). Finally we need three variances 1 (4.9) σ12 (q) := 2 Var(W q + (1 − W )q ), c 1 σ22 (q) := 2 Var(|2W − 1|q ). (4.10) b Wq (1 − W1 )q 1 1 (4.11) |2W2 − 1|q + |2W3 − 1|q − |2W1 − 1|q , σ32 (q) := 2 Var b c c where Wi , i = 1, 2, 3 are iid with the distribution of the cascade generator. It is convenient to define W = e−Y so that the Laplace transform of Y is (4.12)
φ(q) := Ee−qY = E(W q )
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S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
and ar (q) =
21−r φ(rq) . φr (q)
5. Critical Constants We now define the quantity (5.1)
q∗ := sup{q > 0 : a(q) < 1}
so that for q < q∗ we have a(q) < 1. It will turn out that when q < q∗ , the sequence {M (q, l), l ≥ 1} is an L2 -bounded and uniformly integrable martingale and this is the easiest case to analyze. It is always the case that q∗ ≥ 1, which follows from the fact that a(1) =
E(W 2 ) E(W 2 ) = 2E(W 2 ) = 2(E(W ))2 2( 12 )2
so that
1 1 1 1 1 a(1) = 2 Var(W ) + = 2E(W − )2 + ≤ 2 · |1 − |2 + = 1. 4 2 2 2 2 The Mandelbrot-Kahane-Peyri´ ere (MKP) Condition: Let W be the cascade generator and define Wq , q > 0, Xq = EW q so that EXq = 1. The MKP Condition is satisfied for q if (5.2)
E(Xq log2 Xq ) < 1
iff (5.3)
q E(W q log W ) − log E(W q ) < log 2 E(W q )
iff (5.4) Define
q(log φ)′ (q) − log φ < log 2 .
Λ∗ := {q : E(Xq log2 Xq ) < 1}. Then Λ∗ is an interval and we define the second critical constant (5.5)
q ∗ := sup Λ∗ .
Why is q ∗ considered a critical quantity? It turns out that the martingale {M (q, l), l ≥ 1} converges as l → ∞ to M (q, ∞) where ( 0, if q ≥ q ∗ , M (q, ∞) = something non-degenerate, if q < q ∗ . It turns out that the associated martingale is uninformative asymptotically when q > q ∗ . The two critical constants are related numerically by the inequality (5.6)
max(1, q ∗ /2) ≤ q∗ ≤ q ∗ .
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
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To see why the inequality q∗ ≤ q ∗ in (5.6) is true, it suffices to show that if q > 0 satisfies a(q) < 1, then the MKP condition is satisfied for this q. However, if a(q) < 1, then log 2 > log φ(2q) − 2 log φ(q) = log φ(2q) − log φ(q) − log φ(q) Z 2q (log φ)′ (s)ds − log φ(q) = q
and since log φ is convex, (log φ)′ is increasing and the forgoing is bounded below by Z 2q ds − log φ(q) ≥(log φ)′ (q) q
=q(log φ)′ (q) − log φ(q).
We conclude
log 2 > q(log φ)′ (q) − log φ(q) which is equivalent to the MKP condition holding by (5.4). On the other hand, suppose that q∗ < ∞. Since a(q∗ ) = 1, we have in the same way as above log 2 = log φ(2q∗ ) − 2 log φ(q∗ ) = 2 log φ(2q∗ ) − log φ(q∗ ) − log φ(2q∗ ) Z 2q∗ (log φ)′ (s)ds − log φ(2q∗ ) =2 q∗
′
≤2(log φ) (2q∗ )
Z
2q∗
q∗
ds − log φ(2q∗ )
=2q(log φ)′ (2q∗ ) − log φ(2q∗ ).
Therefore, the MKP condition does not hold for 2q∗ , and so q ∗ ≤ 2q∗ .
Example 1. Suppose W is uniformly distributed on [0, 1]. In this case E(W q ) = 1/(1 + q) and so 1 q2 a(q) = 1+ 2 2q + 1 and √ q∗ = 1 + 2 ≈ 2.4. Likewise, q ∗ satisfies the equation q log(1 + q) − = log 2 1+q and so q ∗ ≈ 3.311. Example 2. Suppose more generally that W has the beta distribution with mean 1/2 (i.e., the shape parameters α and β are equal). Then E(W q ) =
(5.7) and q∗ satisfies
Γ(2α)Γ(α + q) Γ(α)Γ(2α + q)
√ 4−α πΓ(α + 2q∗ )Γ(2α + q∗ )2 − Γ(α + 1/2)Γ(2α + 2q∗ )Γ(α + q∗ )2 = 0.
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S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
Example 3. Suppose W has the two point distribution concentrating mass 1/2 at ±p for some 0 ≤ p < 1/2. Then 1 (5.8) E(W q ) = (pq + (1 − p)q ) 2 and 2pq (1 − p)q a(q) = 1 − q . (p + (1 − p)q )2 Note in this case that a(q) ↑ 1 as q ↑ ∞ so q∗ = q ∗ = ∞. Example 4. If W does not have a two point distribution but nevertheless has an atom of size p1 < 1/2 at 1 (and hence by symmetry there is an atom of the same size at 0) we have q ∗ < ∞ (and, hence, also q∗ < ∞). To see this we express the condition (5.3), when q > 1, in the equivalent form E(W q log2 W q ) > 1. (5.9) E(W q ) log2 2E(W q ) Note that if W does not have a two point distribution, then for q > 1, we have P [W q < W ] > 0 and E(W q ) < E(W ) = 1/2, so log2 (2E(wq )) < 0, which explains the sign reversal in (5.9) compared with (5.3). By the dominated convergence theorem the numerator in (5.9) converges to 0 as q → ∞, while the denominator converges to P [W = 1] log2 2P [W = 1] 6= 0. Hence, (5.9) fails for large q. Based on the experience provided in Examples 3 and 4, it natural to wonder how common it can be that q ∗ = ∞. This is discussed in the next proposition.
Proposition 5.1. Unless W has a two point distribution, it must be the case that q ∗ < ∞.
Proof. Because of Example 3, we may assume that W does not have atoms at 0 and 1. Let p ∈ [0, 1/2) be the leftmost point of the support of the distribution of W . Then 1 − p is the rightmost point of the support of the distribution of W . For 0 < ρ < 1, we have θ(ρ) := P [W ≥ ρ(1 − p)] > 0.
Since the distribution of W is not a two point distribution, limρ→1 θ(ρ) < 12 . Thus, we can find and fix a value of 0 < ρ < 1 such that θ(ρ) < 1/2,
0 < ρ < 1.
For this value of ρ, it is convenient to set δ(ρ) := δ = ρ(1 − p).
Apply Jensen’s inequality with the convex function g(x) = x log x, x > 0, to get q q Wq W q W W · 1[W ≥δ] =θ(ρ)E log2 log2 E W ≥ δ c(q) c(q) c(q) c(q) q q W W ≥θ(ρ)E W ≥ δ log2 E W ≥ δ c(q) c(q) Wq W q =E 1 (5.10) W ≥ δ . · log E 2 2E(W q ) [W ≥δ] 2E(W q )
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
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Also we have Wq 1 lim E q→∞ c(q) [W ≥δ]
(5.11)
1 = . 2
To verify (5.11), note that W q) Wq Wq 1 =E =E 1[W ≥δ] + E 1[W 0 log 2 2 2EW q E(W q ) 2 2 ∗ and so the MKP condition (5.3) fails for all large q. Therefore, q < ∞. E
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S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
5.1. Properties of the function ar (q). We need the following properties of the function ar (q). Proposition 5.2. (i) For any fixed r > 1, the function ar (q) (and therefore a(q)) is strictly increasing in q > 0. (ii) For any fixed q > 0, the function log ar (q) is strictly convex in the region r > 1. (iii) If q satisfies the MKP condition, then d log ar (q) < 0, dr r=1 and there exists r0 ∈ (1, 2) such that
(5.15)
ar0 (q) < a1 (q) = 1.
(iv) If the MKP condition fails for q and the inequalities in (5.2), (5.3) or (5.4) are reversed to become strictly greater than, we have d (5.16) log ar (q) > 0, dr r=1 and there exists 0 < r1 < 1 and ar1 (q) < a1 (q) = 1. Proof. (i) Recall the definition of φ from (4.12). For fixed r > 1, if we differentiate with respect to q, we get φ(rq) ′ φr (q)rφ′ (rq) − φ(rq)rφr−1 (q)φ′ (q) = . φr (q) φ2r (q) This is positive iff φ(q)φ′ (rq) > φ(rq)φ′ (q) or φ′ (q) φ′ (rq) > . φ(rq) φ(q) Since r > 1, it suffices to show φ′ /φ is strictly increasing which is true if its derivative is strictly positive. The derivative is φ(q)φ′′ (q) − (φ′ (q))2 φ2 (q) and this is strictly positive iff φ(q)φ′′ (q) > (φ′ (q))2 ,
(5.17) that is iff
2 E(e−qY )E(Y 2 e−qY ) > E(Y e−q/2Y · e−q/2Y )
which follows from the Cauchy–Schwartz inequality. (ii) Fix q > 0 and check that
q 2 ′′ d2 (log a (q)) = φ (rq)φ(rq) − (φ′ (rq))2 r 2 2 dr φ (rq)
which is positive by (5.17). (iii) For fixed q > 0,
d log ar (q) = q(log φ)′ (qr) − log 2 − log φ(q), dr
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
so that
d d dr ar (q) log ar (q) = dr ar (q) r=1
r=1
11
= q(log φ)′ (q) − log φ(q) − log 2 < 0.
Since a1 (q) = 1, we have
d ar (q) < 0, dr r=1 Hence there exists r0 ∈ (1, 2) such that
a1 (q) = 1.
ar0 (q) < a1 (q) = 1.
(iv) If d a (q) d r = dr log ar (q) dr ar (q) r=1
r=1
= q(log φ)′ (q) − log φ(q) − log 2 > 0,
then since log a1 (q) = 0, there exists r1 < 1 such that log ar1 (q) < 0 or ar1 (q) < 1.
6. The Associated Martingale. In this section we study the properties of the process {M (q, l), l ≥ 1} defined in (4.5) for each fixed q > 0. We define the increasing family of σ-fields Fl := σ{W (j|l), j|l ∈ {0, 1}l }
generated by the weights up to and including depth l. Proposition 6.1. For each q > 0, the family
{(M (q, l), Fl ), l ≥ 1}
is a non-negative martingale with constant mean 1 such that M (q, l) converges almost surely to a limiting random variable M (q, ∞): a.s.
M (q, l) → M (q, ∞),
If the MKP condition fails for q, then
E(M (q, ∞) ≤ 1.
P [M (q, ∞) = 0] = 1,
and if q satisfies the MKP condition, then E(M (q, ∞)) = 1 so that P [M (q, ∞) > 0] = 1.
Proof. The martingale property is easily established: X Ql W q (j|i) W q (j|l, jl+1 ) i=1 |Fl E(M (q, l + 1)|Fl ) = E cl c j|l,jl+1
l XY W q (j|i) X q = E W (j|l, j )/c l+1 cl j|l i=1
jl+1
q
=M (q, l)2 · E(W )/c = M (q, l).
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S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
By the martingale convergence theorem (eg, [18], [16]) a non-negative martingale always converges almost surely. The last statements follow by the methods of Kahane and Peyri´ere ([8]). See also Propositions 6.2 and 6.3. Example 5. Recall the example of the two point distribution of Example 3 of Section 4. In this case we have M (q, l) = 1 for all q > 0 and l ≥ 1. For verifying this, the key observation is that W q + (1 − W )q = pq + (1 − p)q .
(6.1)
Recall (5.8) and then observe for l > 1 l l−1 XY XY W q (j|i) W q (j|i) X W q (j|l − 1, jl ) M (q, l) = = c c c j|l i=1
=
X
l−1 Y
j|l−1 i=1
jl
j|l−1 i=1
W q (j|i) c
h
i W q (j|l − 1, 0) + W q (j|l − 1, 1) /c
and since W (j|l − 1, 1) = 1 − W (j|l − 1, 0) we apply (6.1) to get =
l−1 XY W q (j|i) h c
j|l−1 i=1
=
l−1 XY W q (j|i) c
j|l−1 i=1
i pq + (1 − p)q /c
= M (q, l − 1).
One can easily see that M (q, 1) = 1 and the assertion is shown. Define M (q, 0) = 1 and let the martingale differences be d(q, l) := M (q, l) − M (q, l − 1),
l ≥ 1.
For l > 1 we have from the definition of M (q, l) that l−1 i XY W q (j|i) h W q (j|l − 1, 0) + (1 − W (j|l − 1, 0)q d(q, l) = −1 c c j|l−1 i=1
(6.2)
=
l−1 XY W q (j|i) h
j|l−1 i=1
c
i ξ(j|l) .
We now easily see E(d(q, l)|Fl−1 ) = 0. For the conditional variance, note E(ξ(j|l)) = 0 and recall the notation from (4.9) 1 σ12 (q) = Var(ξ(j|l) = 2 Var(W q + (1 − W )q ). c So the conditional variance of d(q, l) is ! l−1 2 X Y q (j|i) W ξ(j|l) |Fl−1 E(d2 (q, l)|Fl−1 ) =E c j|l−1 i=1
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
=
l−1 l−1 X Y W q (j|i) Y W q (p|i) c
j|l−1 i=1 p|l−1
c
i=1
13
E(ξ(j|l)ξ(p|l)).
Since ξ(p|l)) ⊥ ξ(j|l)) if p|l 6= j|l we have 2
E(d (q, l)|Fl−1 ) =
l−1 X Y W q (j|i) 2 c
j|l−1 i=1
=
l−1 XY W 2q (j|i) c(2q)
j|l−1 i=1
σ12 (q)
al−1 (q)σ12 (q)
=M (2q, l − 1)al−1 (q)σ12 (q). Thus the conditional variance of M (q, l) is (6.3)
l X
E(d2 (q, i)|Fi−1 ) =
i=1
i=1
Furthermore, and thus (6.4)
l X
M (2q, i − 1)ai−1 (q)σ12 (q).
E(d2 (q, l)) = E(E(d2 (q, l)|Fl−1 )) = EM (2q, l − 1)al−1 (q)σ12 (q) = al−1 (q)σ12 (q) Var(M (q, l)) =
l X
E(d2 (q, i)) = σ12 (q)
ai−1 (q).
i=1
i=1
This leads to the following facts.
l X
Proposition 6.2. If q < q∗ so that a(q) < 1, the martingale {(M (q, l), Fl ), l ≥ 0} is L2 -bounded and hence uniformly integrable. It follows that (6.5)
E(M (q, ∞) = 1,
M (q, l) = E(M (q, ∞)|Fl ) ,
and M (q, l) → M (q, ∞) almost surely and in L2 . Moreover, if q∗ ≤ q < q ∗ , then the martingale {(M (q, l), Fl ), l ≥ 0} is Lp -bounded for some 1 < p < 2 and, hence, still uniformly integrable, (6.5) still holds and M (q, l) → M (q, ∞) almost surely and in Lp .
Remark. The proof will show that when q∗ ≤ q < q ∗ , we may take p = r0 , where r0 is given in Proposition 5.2 (iii). See (5.15). Proof. Suppose first that q < q∗ . We have from (6.4) that sup E(M (q, l) − 1)2 = sup Var(M (q, l)) l≥0
l≥0
= lim ↑ l→∞
=
∞ X i=1
l X
E(d2 (q, i)
i=1
ai−1 (q)σ12 (q) < ∞.
14
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
The rest follows from standard martingale theory (eg, [16, page 68], [18]). Now let q < q ∗ and we consider uniform integrability without L2 boundedness. Suppose 1 < p ≤ 2 and for two paths j 1 and j 2 denote by mj1 ,j2 the largest i ≤ l such that j 1 |i = j 2 |i. We have p/2 X X Y l 1 q q W (j 1 |i)W (j 2 |i) E E(M (q, l)) = c(q)lp p
j1 |l j 2 |l i=1
X l 1 X ≤ E c(q)lp k=0
l Y i=1
j 1 |l j 2 |l
p/2 W (j 1 |i)W (j 2 |i) q
q
mj 1 ,j 2 =k
X Y k l 1 X W 2q (j|i) · E ≤ lp c (q) j|k i=1
k=0
X
(1) (1) jk+1 ,...,jl (2) (2) jk+1 ,...,jl (1) (2) jk+1 6=jk+1
l Y
W
q
i=k+1
p/2
(1) (1) (2) (2) (j1 , . . . , jk , jk+1 , . . . , jl |i)W q (j1 , . . . , jk , jk+1 , . . . , jl |i)
l
≤
1 X X pq k E W · clp (q) k=0 j|k
E
≤
X
k=0
cp (q)
E
W
q
c(l−k)p (q)
·
(1) (1) jk+1 ,...,jl (2) (2) jk+1 ,...,jl (2) (1) jk+1 6=jk+1
l X c(pq) k
(1)
X
l Y
i=k+1
1
(1)
jk+1 ,...,jl
i=k+1
(2) (2) jk+1 ,...,jl (1) (2) jk+1 6=jk+1
=
l X k=0
=
k
ap (q)
l Y
1 c(l−k)p (q)
E(W (1 − W ))q
p/2
(2) (2) (1) (1) (j1 , . . . , jk , jk+1 , . . . , jl |i)W q (j1 , . . . , jk , jk+1 , . . . , jl |i)
p/2 (2) (2) (1) (1) W q (j1 , . . . , jk , jk+1 , . . . , jl |i)W q (j1 , . . . , jk , jk+1 , . . . , jl |i)
q
q 2(l−k−1) 2(l−k−1)
E(W (1 − W )) (EW )
p/2 2p/2 c(q)
X l k=0
ap (q)k .
2
p/2 ·2
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
15
Here a product over the empty set is equal to 1. By Proposition 5.2 (iii) there is a p ∈ (1, 2) such that ap (q) < 1. For this p the martingale {(M (q, l), Fl ), l ≥ 0} is Lp -bounded, and the rest follows, once again, from standard martingale theory. 6.1. The distribution of M (q, ∞). The distribution of M (q, ∞) satisfies a simple recursion which can be used to derive additional information. Proposition 6.3. Suppose {M (q, ∞), M1 (q, ∞), M2 (q, ∞)} are iid with the same distribution as M (q, ∞), the martingale limit. Let W have the distribution of the cascade generator and suppose W and {M (q, ∞), M1 (q, ∞), M2 (q, ∞)} are independent. Then d
M (q, ∞) = W q
(6.6)
M2 (q, ∞) M1 (q, ∞) + (1 − W )q c(q) c(q)
and for any q > 0, (6.7)
P [M (q, ∞) = 0] = 0 or 1,
so that E(M (q, ∞)) = 1 implies P [M (q, ∞) = 0] = 0. Proof. We write M (q, ∞) = lim
l→∞
l XY W q (j|i)
cl
j|l i=1
l l X Y X Y W q (1, j2 , . . . , jl ) W q (0, j2 , . . . , jl ) + = lim l→∞ cl cl j2 ,...,jl i=1
j2 ,...,jl i=1
X = lim W q (0) l→∞
l Y
j2 ,...,jl i=2
l X Y W q (1, j2 , . . . , jl ) W q (0, j2 , . . . , jl ) q + (1 − W (0)) cl cl j2 ,...,jl i=2
M1 (q, ∞) M2 (q, ∞) + (1 − W (0))q . c c Now we verify (6.7). Define d
=W q (0)
p0 =P [M (q, ∞) = 0]
pW (0) =P [W = 0] = P [W = 1]. Then since c(q) 6= 0
p0 =P [M (q, ∞) = 0] = P [W q M1 (q, ∞) + (1 − W )q M2 (q, ∞) = 0] =P [′′ , W = 0] + P [′′ , W = 1] + P [′′ , 0 < W < 1].
¿From this we conclude so that
p0 = 2pW (0)p0 + (1 − 2pW (0))p20
p0 (1 − 2pW (0)) = p20 (1 − 2pW (0)). If 0 < pW (0) < 1/2, then p0 = p20 and p0 = 0 or 1. If pW (0) = 1/2, then P [W = 0] = P [W = 1] = 1/2 and W has a two point distribution and hence from Example 5 we know M (q, l) = 1 which implies M (q, ∞) = 1.
16
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
7. Estimation: Subcritical Consistency We propose two estimators of the structure function which depend on scaled summed powers of the wavelet coefficients {Z(q, l), l ≥ 1}. These are P l −1 log2 2n=0 |d−l,n |q − ql/2 log2 Z(q, l) (7.1) = τˆ1 (q) = τˆ1 (q, l) = l l ! P2l+1 −1 q |d | Z(q, l + 1) −(l+1),n n=0 τˆ2 (q) = τˆ2 (q, l) = log2 (7.2) . = log2 P2l −1 q/2 Z(q, l) 2 |d−l,n |q n=0
Analysis depends on showing that scaled versions of Z(q, l) are well-approximated by the martingale and this is discussed next. Recall notational definitions (4.1), (4.2), (4.3), (4.6). Proposition 7.1. For q > 0, Z(q, l) P − M (q, l) → 0. l cb If q = 6 q ∗ , the convergence is almost sure and if q < q∗ , the convergence is in L2 . Thus Z(q, l) → M (q, ∞) cl b in the appropriate sense, depending on the case. (7.3)
Proof. Begin by writing l i XY Z(q, l) W q (j|i) h |2W (j|l, 0) − 1|q − M (q, l) = − 1 cl b c b j|l i=1
(7.4)
=
l XY W q (j|i)
c
j|l i=1
ξ(j|l, 0)
where ξ(j|l, 0) ⊥ ξ(p|l, 0) if j|l 6= p|l. Also Eξ(j|l, 0) = 0 and recall the notation from (4.10)
1 Var(|2W − 1|q ). b2 If q < q∗ , so a(q) < 1, then similar to the calculations leading to (6.3) and (6.4) we find ! l l 2 X X Z(q, l) Y W q (j|i) Y W q (p|i) − M (q, l) = ξ(j|l, 0)ξ(p|l, 0) E E cl b c c σ22 (q) := Eξ 2 (j|l, 0) =
j|l p|l
=σ22 (q)
i=1
(2EW 2q )l c2l (q)
i=1
= σ22 (q)al (q)
→0 as l → ∞ since a(q) < 1. This shows the L2 –convergence. For q > 0, the same method shows Z(q, l) 2 |Fl =σ22 (q)M (2q, l)al (q) − M (q, l) E cl b
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
=σ22 (q)
(7.5)
l XY W 2q (j|i) j|l i=1
c2 (q)
17
=: σ22 (q)V (q, l),
and we need to show V (q, l) → 0, almost surely as l → ∞. If the MKP condition fails, then M (q, l) → 0 as l → ∞ and 2 V (q, l) ≤ M (q, l) → 0. If the MKP condition holds, then from Proposition 5.2 (iii), there exists r0 ∈ (1, 2) such that ar0 (q) < 1, and for p = r0 /2 ∈ (1/2, 1) we have by the triangle inequality (7.6)
0 ≤ V (q, l)p ≤
l XY W 2pq (j|i) j|l i=1
c2p (q)
a.s.
= (ar0 (q))l M (r0 q, l) → 0,
as l → ∞, since M (r0 q, ∞) < ∞ almost surely. So in all cases V (q, l) → 0. For any δ > 0, ǫ > 0 we have Z(q, l) Z(q, l) − M (q, l) > ǫ|Fl ] =P [ l − M (q, l) > ǫ|Fl ]1[V (q,l)σ22 (q)>δ] P [ l cb cb Z(q, l) + P [ l − M (q, l) > ǫ|Fl ]1[V (q,l)σ22 (q)≤δ] cb ! Z(q, l) 2 ≤1[V (q,l)σ22 (q)>δ] + ǫ−2 E − M (q, l) |Fl 1[V (q,l)σ22 (q)≤δ] cl b =1[V (q,l)σ22 (q)>δ] + ǫ−2 V (q, l)σ22 (q)1[V (q,l)σ22 (q)≤δ] ≤1[V (q,l)σ22 (q)>δ] +
δ2 . ǫ2
a.s.
Take expectations and use V (q, l) → 0 and the arbitrariness of δ to conclude Z(q, l) P − M (q, l) → 0, l cb
as l → ∞. For almost sure convergence, when q < q ∗ , we get from (7.6) that l V (q, l) ≤ ar0 (q)1/p M 1/p (r0 q, l) P and so l V (q, l) < ∞ almost surely. Thus, for any ǫ > 0, ! Z(q, l) 2 X X X Z(q, l) E −M (q, l) |F = (const) V (q, l) < ∞, P [| l −M (q, l)| > ǫ|Fl ] ≤ ǫ−2 l cb cl b l
l
and by a generalization of the Borel-Cantelli lemma ([16, page 152]) we have
Z(q, l) a.s. − M (q, l) → 0. cl b For q > q ∗ , we prove almost sure convergence from Proposition 5.2 (iv) in a similar way. We use this comparison result Proposition 7.1 to get consistent estimators of the structure function τ (q) in the subcritical case, by which we mean the case where the MKP condition holds.
18
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
Proposition 7.2. Define τˆi (q) for i = 1, 2 by (7.1) and (7.2). Provided q < q ∗ , so that the MKP condition holds, both estimators are almost surely consistent for τ (q): a.s.
τˆi (q) → τ (q),
i = 1, 2,
as l → ∞. Proof. In (7.3), take logarithms to the base 2 to get (7.7)
log2 Z(q, l) − l log2 c(q) − log2 b → log2 M (q, ∞),
almost surely as l → ∞. Divide through by l to get consistency of τˆ1 (q). To get the consistency of τˆ2 (q), note from (7.7) that log2 Z(q, l + 1) − log2 Z(q, l) − (l + 1 − l)τ (q) → 0 almost surely which proves consistency of τˆ2 (q). 8. Subcritical Asymptotic Normality of Estimators. In this section we discuss second order properties of the estimators τˆi (q), i = 1, 2, defined in (7.1) and (7.2). The asymptotic normality for τˆ1 (q) requires a bias term which cannot be eliminated. This drawback, is eliminated by using τˆ2 (q), whose definition in terms of differencing removes the bias term. However, take note of the suggestive remarks at the end of this Section 8 about mean squared error. For this section it is convenient to write E Fl and P Fl for the conditional expectation and conditional probability with respect to the σ-field Fl . 8.1. Asymptotic Normality of τˆ1 (q). Begin by writing (8.1) (8.2)
l i XY W q (j|i) h |2W (j|l, 0) − 1|q Z(q, l) − M (q, l) = − 1 cl b c b j|l i=1 X =: Z(j|l) j|l
where E Fl (Z(j|l)) =0 Y l W 2q (j|i) l 2 Fl E (Z(j|l)) = a (q)σ22 (q), c(2q) i=1
and recall σ22 (q) is defined in (4.10). Therefore, X (8.3) E Fl (Z(j|l))2 = M (2q, l)al (q)σ22 (q). j|l
Our strategy for the central limit theorem is to regard Z(q,l) −M (q, l) as a sum of random variables cl b which are conditionally independent given Fl and then apply the Liapunov condition ([18]) for asymptotic normality in a triangular array.
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
Proposition 8.1. If 2q < q ∗ , then as l → ∞ # " Z(q,l) − M (q, l) l (8.4) ≤ x → P [N (0, 1) ≤ x] P Fl p c b M (2q, l)al (q)σ22 (q)
19
a.s.
where N (0, 1) is a standard normal random variable with mean 0 and variance 1. Taking expectations in (8.4) yields # " Z(q,l) − M (q, l) lb c (8.5) ≤ x → P [N (0, 1) ≤ x]. P p M (2q, l)al (q)σ22 (q) Proof. By Proposition 5.2 (iii) there is δ > 0 such that both 2q + δ < q ∗ and
(8.6)
a1+δ/2 (2q) < 1.
Asymptotic normality in (8.4) will be shown if we establish the Liapunov condition P 2+δ Fl j|l E |Z(j|l)| (8.7) →0 a.s., (M (2q, l)al (q))(2+δ)/2 where the denominator comes from (8.3). The numerator in the left side of (8.7) is bounded above by 2+δ l 2+δ |2W (j|l, 0) − 1|q q (j|i) X Y W − 1 E Fl c b j|l
i=1
=c1
l XY W q(2+δ) (j|i) cl ((2 + δ)q) j|l
i=1
c((2 + δ)q)
cl(2+δ) (q)
=c1 M ((2 + δ)q, l)(a2+δ (q))l , where
|2W (j|l, 0) − 1|q 2+δ c1 = E − 1 . b So the ratio in (8.7), apart from constants, is bounded by
M ((2 + δ)q, ∞)(a2+δ (q))l M ((2 + δ)q, l)(a2+δ (q))l ∼ . M (2q, l)1+δ/2 (a2 (q))(1+δ/2)l M (2q, ∞)1+δ/2 (a2 (q))(1+δ/2)l
Note that the two random variables M ((2 + δ)q, ∞) and M (2q, ∞) are non zero with probability 1 by Proposition 6.1. Check that a2+δ (q) = a1+δ/2 (2q) < 1. (a2 (q))1+δ/2 So the Liapunov ratio is asymptotic to a finite nonzero random variable times (a1+δ/2 (2q))l where a1+δ/2 (2q) < 1 and the result is proven. Remark 8.1. In the denominator of (8.5) we may replace M (2q, l) by its limit M (2q, ∞). This follows since almost surely 0 < M (2q, ∞) < ∞ for 2q < q ∗ and thus s ! Z(q,l) − M (q, l) M (2q, l) l p cb , ⇒ (N (0, 1), 1) M (2q, ∞) M (2q, l)al (q)σ22 (q)
20
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
by [2]. The desired result is obtained by multiplying components. Remark 8.2. Set Z(q,l) − M (q, l) l . Nl := p c b M (2q, l)al (q)σ22 (q)
(8.8) Then in R2 , as l → ∞
(Nl , Nl+1 ) ⇒ N1 (0, 1), N2 (0, 1) , where Ni (0, 1), i = 1, 2 are iid standard normal random variables. To see this, write for any x, y ∈ R
P [Nl ≤ x, Nl+1 ≤ y] =EP Fl+1 [Nl ≤ x, Nl+1 ≤ y] =E1[Nl ≤x] P Fl+1 [Nl+1 ≤ y].
By Proposition 8.1,
P Fl+1 [Nl+1 ≤ y] = Φ(y) + ǫl (y)
a.s.
L1
where Φ(y) is the standard normal cdf and where ǫl (y) → 0 and |ǫl (y)| ≤ 2. So P [Nl ≤ x, Nl+1 ≤ y] =E1[Nl ≤x] Φ(y) + ǫl (y) =E1[Nl ≤x] Φ(y) + o(1)
from the dominated convergence theorem, and hence we get convergence to →Φ(x)Φ(y). We now describe how this central limit behavior transfers to τˆ1 (q). Corollary 8.1. Under the assumptions in force in Proposition 8.1, we have τˆ1 (q) − τ (q) − l−1 log2 bM (q, l) √ (8.9) ⇒ N (0, 1). 2 l M (2q,∞)a (q)σ2 (q) l log 2·M (q,l)
Remark. The bias term l−1 log2 bM (q, l) cannot be neglected. Proof. For brevity, write
(8.10)
d(q) := M (2q, ∞)al (q)σ22 (q),
and using the notation of (8.8) we have
Since
p Z(q, l) = cl b Nl d(q) + M (q, l) .
1 log2 Z(q, l), l p lˆ τ1 (q) = l log2 c + log2 b + log2 Nl d(q) + M (q, l) τˆ1 (q) =
we have and thus
p Nl d(q) . l τˆ1 (q) − τ (q) = log2 bM (q, l) + log2 1 + M (q, l)
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
21
∗ Since p by (5.6) and assumption 2q < q we have q < q∗ , we know that d(q) → 0. Therefore, Nl d(q)/M (q, l) → 0, and the desired result follows by using the relation log(1 + x) ∼ x for x ↓ 0.
The bias term in (8.9) is an unpleasant feature and thus we consider how to remove it by differencing. 8.2. Asymptotic Normality of τˆ2 (q). We now consider the asymptotic normality of τˆ2 (q). It is possible to proceed from Proposition 8.1 but it turns out to be simpler to proceed with a direct proof. Proposition 8.2. Suppose 2q < q ∗ . Then √
(8.11)
τˆ2 (q) − τ (q)
M (2q,∞)al (q)σ32 (q) log 2·M (q,∞)
⇒ N (0, 1),
where σ32 (q) is defined in (4.11). Proof. Begin by observing that Z(q, l + 1) Z(q, l) − cl+1 b cl b " q l X Y W q (j|i) W q (j|l, 1) |2W (j|l, 1, 0) − 1|q W (j|l, 0) |2W (j|l, 0, 0) − 1|q = + c c b c b j|l i=1 # |2W (j|l, 0) − 1|q − b l X Y W q (j|i) H(j|l) =: c j|l
i=1
where we have set (8.12) H(j|l) =
W q (j|l, 1) |2W (j|l, 1, 0) − 1|q |2W (j|l, 0) − 1|q W q (j|l, 0) |2W (j|l, 0, 0) − 1|q + − . c b c b b
Check that 1 1 E Fl H(j|l) = + − 1 = 0 2 2 and E Fl H 2 (j|l) =σ32 (q). It follows that conditionally on Fl we may treat Z(q, l + 1) Z(q, l) − cl+1 b cl b
22
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
as a sum of iid random variables with (conditional) variance 2 l Z(q, l) 2 X Fl Y W q (j|i) Fl Z(q, l + 1) H(j|l) E = E − cl+1 b cl b c(q) i=1
j|l
=
l XY W 2q (j|i) l a (q)EH 2 (j|l) c(2q) j|l
i=1
=M (2q, l)al (q)σ32 (q).
As in the proof of Proposition 8.1 we may check the Liapunov condition and conclude Z(q,l) cl b M (2q, ∞)al (q)σ32 (q) Z(q,l+1) cl+1 b
p
or equivalently
−
⇒ N (0, 1),
Z(q,l) − 1 c−1 Z(q,l+1) Z(q,l) cl b p ⇒ N (0, 1), M (2q, ∞)al (q)σ32 (q)
and since Z(q, l)/(cl b) → M (q, ∞) we have c−1 Z(q,l+1) − 1 M (q, ∞) Z(q,l) p (8.13) ⇒ N (0, 1). M (2q, ∞)al (q)σ32 (q) Since
c−1
Z(q, l + 1) P − 1 → 0, Z(q, l)
it follows that Z(q, l + 1) τˆ2 (q) − τ (q) = log2 c−1 Z(q, l) log 1 + c−1 Z(q,l+1) − 1 Z(q,l) = log 2 ∼
c−1 Z(q,l+1) Z(q,l) − 1 log 2
in probability. Combine this with (8.13) to complete the proof. For statistical purposes, the result (8.11) contains unobservables so as in [20, 17], consideration needs to be given to replacing quantities which are not observed by observable estimators. We assume that the random measure µ∞ is observed, or equivalently that the wavelet coefficients {d−l,n } are known. This means we have the quantities {Z(q, l)}. We define the following useful observable quantity " l XY W q (j|l, 1)|2W (j|l, 1, 0) − 1|q W q (j|l, 0)|2W (j|l, 0, 0) − 1|q + W 2q (j|i) D 2 (q, l) = Z(q, l + 1) Z(q, l + 1) j|l i=1
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
|2W (j|l, 0) − 1|q − Z(q, l)
(8.14) =:
l XY
23
#2
W 2q (j|i)V 2 (j|l)
j|l i=1
=:
l XY j|l i=1
"
C A+B W 2q (j|i) − Z(q, l + 1) Z(q, l)
#2
.
Note that in this notation, A+B C − , cb b 2 where H(j|l) is defined in (8.12). Recall also that EH (j|l) = σ32 (q). In terms of the wavelet coefficients we have (8.15) #2 " q 2−q(l+1)/2 q 2−ql/2 X |d(−l, (j|l, 0))|q 2−q(l+1)/2 |d(−l, (j|l, 1))| |d(−l, (j|l)| , + − D 2 (q, l) = Z(q, l + 1) Z(q, l + 1) Z(q, l) H(j|l) =
j|l
showing that D 2 (q, l) is an observable statistic. Corollary 8.2. Suppose 2q < q ∗ . Then τˆ2 (q) − τ (q) ⇒ N (0, 1) D(q, l)/ log 2
(8.16) as l → ∞.
Proof. Because of (8.11), it suffices to show D 2 (q, l) P → 1, M (2q, ∞)al (q)σ32 (q)/M 2 (q, ∞)
as l → ∞. This is equivalent to showing
Z 2 (q,l) D 2 (q, l) P c2l (q)b2 (q) → M (2q, l)al (q)σ32 (q)
1.
After some simple algebra, this ratio is the same as ! #2 P Ql W 2q (j|i) " C Z(q, l)/cl b A+B j|l i=1 c(2q) − . bc Z(q, l + 1)/cl+1 b b M (2q, l)σ32 (q) Since M (2q, l) → M (2q, ∞), it suffices to show that the numerator converges in probability to M (2q, ∞)σ32 (q). Due to (7.3), we write the numerator as ! l A + B 2 A + B C XY W 2q (j|i) h A + B C i2 + op (1)2 + op (1)2 − − c(2q) bc b bc b bc j|l i=1
=I + II + III.
24
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
q = 0.25, shape = 1 0.10
q = 0.75, shape = 1
0.05
0.08
-0.05
0.06 0.04
10 12 14 16
tau2 0.0
tau1 0.10
0.12
0.14
10 12 14 16
-2
-1 0 1 Quantiles of standard normal
2
-2
-1 0 1 Quantiles of standard normal
2
Figure 1. Normal QQ-plots of τˆ1 (q) − τ (q) (left) and τˆ2 (q) − τ (q) (right). As in Theorem 3.5 of [17], A + B
C 2 = M (2q, ∞)σ32 (q), bc b as desired. The terms I and II can readily be shown to go to 0. I → M (2q, ∞)E
−
Figure 1 shows normal QQ plots of τˆi (q) − τ (q), i = 1, 2, from simulated cascade data with beta distributed cascade generator with shape parameter 1 (this makes the distribution uniform). The left plot is for q = 0.75 and the right is for q = 0.25. Each plot presents 4 graphs as the depth l increases to 16. Note the better agreement of τˆ2 (q) to normality compared with τˆ1 (q). Concluding remarks on mean squared error: Examining Corollary 8.1 and Proposition 8.2 yields that in the region 2q < q ∗ , the conditional mean squared error of τˆ1 (q) is of the form (1)
(2)
Op (al (q))2 Op (1) + l2 l2 (3)
while that of τˆ2 (q) is Op
2 al (q) .
9. Supercritical Asymptotics; Lack of Consistency A critical issue with both the wavelet based estimator and the moment based ones used in [17], is that the asymptotic properties of the estimators are only valid in a certain range of q-values. For the wavelet estimators, we require q < q ∗ for consistency and for the asymptotic normality results we require 2q < q ∗ . We now show that the range q > q ∗ is uninformative for our estimators and in fact our estimators are misleading when extended to inference for values beyond q ∗ . A reliable estimate of q ∗ would be valuable information. In place of such an estimate it is likely that a graphical procedure is possible based on the following.
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
25
0.0 0.2 0.4 0.6 0.8 1.0 1.2
q = 8 Z/max.Z shape = 10
10 12 14 16 18
2
4
6
8
10
Figure 2. Density plots of Z(q, l)/Z ∨ (q, l). Let τˆi∨ (q) (i = 1, 2) have the same definition as τˆi (q) except that sum is replaced by max. Thus we can define by analogy with (4.3) ∨
Z (q, l) =
l _Y j|l i=1
W (j|i)q |2W (j|l, 0) − 1|q .
Note that q Z ∨ (q, l) = Z ∨ (1, l) .
For large values of q, namely for q ≥ q ∗ , Z(q, l) is sufficiently well approximated by its largest summand Z ∨ (q, l). Figure 2 presents a density plot of simulated values of Z(q, l)/Z ∨ (q, l) as the depth l increases from 10 to 18; note the densities concentrate most mass around the point 1. The cascade generator is a beta distribution with shape parameter 10. Based on the idea of approximating Z(q, l) by Z ∨ (q, l), since log Z ∨ (q, l) = q log Z ∨ (1, l) is linear in q, we anticipate that τˆ1 (q) should also be linear in q rendering τˆ1 (q) largely uninformative for inference purposes in the q ≥ q ∗ –region. A rough estimate of q ∗ would be provided by the q-value where the plots of τˆ1 (q) starts to look linear. Computer simulations offer strong support for these remarks. Figure 3 shows overlaid simulated values for τˆi (q), τˆi∨ (q), i = 1, 2 for large values of q. In the range of q–values beyond q ∗ ≈ 3.3, it is remarkable how linear the plots for τˆ1 (q) and τˆ1∨ (q) look and also how closely τˆ1∨ (q) approximates τˆ1 (q). Note the values in the plots have been multiplied by -1 to make the plots increasing and the cascade generator is a beta distribution with shape parameter 1. We now assume that q ∗ < ∞ and examine this supercritical phenomenon when q ≥ q ∗ in more detail. We will prove the asymptotic linearity of the estimator τˆ1 (q) for q ≥ q ∗ . In particular, the estimator τˆ1 (q) is not consistent when q > q ∗ , and neither is the estimator τˆ2 (q).
26
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
4
levels = 16, q.step = 0.5, shape = 1
3
tau.1 tau.2 theory
-1
0
1
tau(q) 2
max.1 max.2
0
2
4
6
8
10
q Figure 3. Plots of τˆi (q), τˆi∨ (q) for q∗ ≈ 2.4. We start by introducing new notation. Let (9.1)
l
U (q, l) = c(q) M (q, l) =
l XY
W (j|i)q ,
j|l i=1
(9.2) and define, for q > 0, (9.3)
U ∗ (l) = max j|l
l Y
W (j|i),
i=1
l→∞
m(q) = lim inf l→∞
l ≥ 1,
1 log2 U (q, l) , l
m(q) = lim sup
1 log2 U (q, l) l
as well as (9.4)
m∗ = lim sup l→∞
m∗ = lim inf l→∞
q > 0, l ≥ 1 ,
1 log2 U ∗ (l) , l 1 log2 U ∗ (l) . l
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
27
It is immediate that for all q > 0 and 0 ≤ θ ≤ q (9.5)
(U ∗ (l))q ≤ U (q, l) ≤ (U ∗ (l))θ U (q − θ, l) ≤ 2l (U ∗ (l))q .
In particular, for every q > 0
m(q) − 1 ≤ qm∗ ≤ m(q) ,
(9.6)
m(q) − 1 ≤ qm∗ ≤ m(q)
almost surely. Note that it follows from Proposition 6.1 that for 0 < q < q ∗ (9.7)
m(q) = m(q) = τ (q) .
Since by the triangle inequality for all 0 < ρ < 1 and q > 0 (U (q, l))ρ ≤ U (ρq, l) ,
(9.8) we see that (9.9)
m(ρq) ≥ ρm(q) .
For a q ≥ q ∗ and 0 < ρ < q ∗ /q we hence get 1 1 m(q) ≤ m(ρq) = τ (ρq) , ρ ρ and letting ρ ↑ q ∗ /q we conclude that for every q ≥ q ∗ (9.10)
m(q) ≤ q
τ (q ∗ ) . q∗
On the other hand, it follows from (9.5) that for all q > 0 and 0 ≤ θ ≤ q m(q) ≤ θm∗ + m(q − θ) .
Using (9.6) we obtain from here
m(q2 ) + m(q1 − θ) q2 for all q1 , q2 > 0 and 0 ≤ θ ≤ q1 . In particular, if 0 < q1 < q ∗ , then for every 0 < q3 < q1 we choose θ = q1 − q3 and conclude, using (9.7), that m(q1 ) ≤ θ
Therefore, for all q > 0
m(q1 ) − m(q3 ) τ (q1 ) − τ (q3 ) m(q2 ) ≥ = . q2 q1 − q3 q1 − q3 m(q) ≥ sup τ ′ (p) . q 0 0 .
Let 0 < ǫ < 1. Note that it follows from (9.14) that for all l large enough, 1 ∗ ∗ (9.17) P U ∗ (l) ≥ 2(1−ǫ)lτ (q )/q ≥ . 2 For l ≥ 1 let n o Nl = card j|l : W (j|i) ≥ θ for all i = 1, . . . , l .
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
29
By definition N0 = 1. Observe that for all l ≥ 0 Nl+1 = Nl + Ml , where, given N0 , N1 , . . . , Nl , the distribution of Ml is Binomial with parameters Nl and p2 . Therefore, (Nl ) is a supercritical branching process with progeny mean m = 1 + p2 > 1 and extinction probability 0. By Theorem I.10.3 of [1], page 30, lim
(9.18)
l→∞
Nl ˆ > 0 a.s. . =N (1 + p2 )l
Let now 0 < δ < 1. It follows by the definition of (Nl ) that for every l ≥ 1 q (l) Z(l, q) ≥ θ [δl]q max Uk∗ l − [δl] |2Wk − 1|q , (9.19) k=1,... ,N[δl]
where
and
Uk∗ l − [δl] , k ≥ 1 are iid with the law of U ∗ l − [δl] (l)
(Wk , k ≥ 1) are iid with the law of W .
The two sequences are independent, and also independent of N[δl] . All the random variables defined above can be assumed to be defined, for all l and k, on the same probability space (Ω, F, P ). We introduce several events. Let d = (1 + p2 )1/2 > 1. Put n o Ω1 = Nl ≥ dl for all l large enough . It follows from (9.18) that P (Ω1 ) = 1. Let, further, (l) Ω2
=
[δl] d[ n
k=1
o ∗ ∗ (l) |2Wk − 1| ≥ θ and Uk∗ l − [δl] ≥ 2(1−ǫ)(l−[δl])τ (q )/q , (l)
l ≥ 1. Note that by (9.17) we have P (Ω2 ) ≥ 1 − e−cδl for some c > 0 and all l ≥ 1, and so letting (l)
Ω2 = lim inf Ω2 , l→∞
we see by Borel-Cantelli lemma that P (Ω2 ) = 1. Therefore, P (Ω1 ∩ Ω2 ) = 1 as well. However, for every ω ∈ Ω1 ∩ Ω2 we have by (9.19) Z(l, q) ≥ θ [δl]q 2q(1−ǫ)(l−[δl])τ (q
∗ )/q ∗
θq
for all l large enough, which implies that mZ (q) ≥ qδ log2 θ + (1 − ǫ)(1 − δ)q Letting δ → 0 and ǫ → 0 we conclude that (9.20)
mZ (q) ≥ q
τ (q ∗ ) . q∗
Now the statement (9.15) follows from (9.16) and (9.20).
τ (q ∗ ) a.s. . q∗
30
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
Finally, it follows from part (iv) of Proposition 5.2 that τ (q) > q
τ (q ∗ ) q∗
for all q > q ∗ . Hence, the estimator τˆ1 (q, l) is not a consistent estimator of τ (q) if q > q ∗ . Here is an immediate corollary. Corollary 9.1. The estimator τˆ2 (q, l) is not a (strongly) consistent estimator of τ (q) if q > q ∗ . Proof. Notice that for every l ≥ 1,
l−1
τˆ1 (q, l) =
1X τˆ2 (q, j) , l j=0
where Z(q, 0) = 1. Therefore, if for some q > q ∗ τˆ2 (q, l) → τ (q) a.s. as l → ∞, then so does τˆ1 (q, l), which contradicts Theorem 9.1. An estimator related to τˆ2 (q, l) is τˆ3 (q, l) = log2 Since
U (q, l + 1) U (q, l)
:= log2 R(q, l), l ≥ 1 . l−1
(9.21)
1X 1 τˆ3 (q, j) , log2 U (q, l) = l l j=0
where U (q, 0) = 1, (9.13) and the same argument as that of Corollary 9.1 shows that τˆ3 (q, l) is not a strongly consistent estimator of τ (q) if q > q ∗ (even though it is a strongly consistent estimator of τ (q) if q < q ∗ ). We can say more, however. Note that 0 ≤ R(q, l) ≤ 2 for all q and l. Furthermore, ER(q, l) = c(q) = 2τ (q) for all q and l . Therefore, if for some q > q ∗ , τˆ3 (q, l) converges a.s. to some limit τ3 (q) as l → ∞, then the 2τ3 (q) must have a finite expectation equal to 2τ (q) . On the other hand, by (9.13) and (9.21) we must have τ3 (q) equal to qτ (q ∗ )/q ∗ a.s.. This contradiction shows that τˆ3 (q, l) cannot converge a.s. as l → ∞ if q > q ∗ . We conjecture that the same is true for τˆ2 (q, l), in the sense that it does not converge a.s. as l → ∞ if q > q ∗ . A possibility is that τˆ2 (q, l) converges in probability, and is weakly consistent for q > q ∗ . Whether or not this is true remains an open question. 10. Concluding Remarks While Ossiander and Waymire’s estimator for τ (q) is consistent for random cascades, we also check empirically by simulation that it is an appropriate time domain method for conservative cascades. By time domain estimator we mean ! X 1 |µ∞ (I(j|l))|q . τˆtime (q) = log2 l j|l
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
31
levels = 16, q.step = 0.1, instances = 50, shape = 1
-1.0
-0.5
tau(q) 0.0 0.5
1.0
tau.1 tau.2 theory time.one
0.0
0.5
1.0
1.5 q
2.0
2.5
3.0
Figure 4. Plots of the two wavelet estimators and the time domain estimator for τ (q) with q < q∗ ≈ 3.3. We show in Figure 4 that the time domain estimator gives equally good results compared with the two wavelet estimators. The cascade generator is a beta distribution with shape parameter 1. The plot is for q values below q ∗ ≈ 3.3. Note that the τ values are multiplied by -1. One of the advantages of the wavelet method is its ability to filter deterministic trends because different wavelet families have different vanishing moments; i.e., they are orthogonal to low degree polynomials. The Haar wavelets are “blind” to additive constants. Figure 5 illustrates the failure of the time domain method to cope with the presence of an additive constant. The cascade is generated with a beta distribution of shape parameter 1 then a fixed constant 0.1 is added to the cascade. The wavelet estimators give the same values regardless of the presence of the additive constant. Do our wavelet methods work with other wavelet families. One reason for using other wavelets is that the Haar wavelets have only one vanishing moment and can remove only an additive constant. Other wavelet families with higher vanishing moments can remove higher degree deterministic trends. Empirical simulated evidence (shown in Figure 6) suggests that other wavelets do indeed work. Figure 6 shows that τˆ1 works quite well in the case of the D4 wavelet and the presence of an additive linear trend. Note that the D4 wavelet has four vanishing moments (i.e., it is blind to cubic polynomials). The time domain method performs poorly as does τˆ2 . Theoretical investigations are necessary to confirm the validity of the wavelet method for wavelets other than the Haar.
32
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
tau(q) 0.0 0.5
1.0
levels = 16, q.step = 0.1, instances = 50, shape = 1, trend added
-1.0
-0.5
tau.1 tau.2 theory time.trend.one
0.0
0.5
1.0
1.5 q
2.0
2.5
3.0
Figure 5. Plots of the wavelet and time domain estimators for a cascade with an additive constant.
References [1] K. Athreya and P. Ney. Branching Processes. Springer-Verlag, New York, 1972. [2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. [3] A. Feldmann, A. C. Gilbert, and W. Willinger. Data networks as cascades: Investigating the multifractal nature of Internet WAN traffic. In Proc. of the ACM/SIGCOMM’98, pages 25–38, Vancouver, B.C., 1998. [4] U. Frisch and G. Parisi. Fully developed turbulence and intermittancy. In M. Ghil, editor, Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland, Amsterdam, 1985. [5] A.C. Gilbert, W. Willinger, and A. Feldmann. Scaling analysis of conservative cascades, with applications to network traffic. IEEE Transactions on Information Theory, 45(3):971–991, 1999. [6] V. K. Gupta and E. C. Waymire. A statistical analysis of mesoscale rainfall as a random cascade. Journal of Applied Meteorology, 32:251–267, 1993. [7] Richard Holley and Edward C. Waymire. Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Probab., 2(4):819–845, 1992. [8] J.P. Kahane and J. Peyri´ere. Sur certaines martingales de b. mandelbrot. Advances in Mathematics, 22:131– 145, 1976. [9] J.F.C. Kingman. The first birth problem for an age–dependent branching process. The Annals of Probability, 3:790–801, 1975. [10] A.N. Kolmogorov. Local structure of turbulence in an incompressible liquid for very large Reynolds numbers. C.R. Doklady Acad. Sci. URSS (N.S.), 30:299–303, 1941. [11] H. Mahmood. Evolution of Random Search Trees. Wiley, New York, 1992.
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
33
levels = 16, q.step = 0.1, instances = 50, shape = 1, trend added, D4
-1
0
tau(q)
1
2
tau.1 tau.2 theory time.trend.one
0.0
0.5
1.0
1.5 q
2.0
2.5
3.0
Figure 6. Plots of the wavelet and time domain estimators using D4 wavelets and in the presence of an additive linear trend.
[12] B. B. Mandelbrot. Intermittant turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. Journal of Fluid Mechanics, 62:331–358, 1974. [13] B. B. Mandelbrot. Limit Lognormal Multifractal Measures. In Gotsman, Ne’eman, and Voronel, editors, The Landau Memorial Conference, pages 309–340, Tel Aviv, 1990. [14] B.B. Mandelbrot. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer-Verlag, New York, 1998. [15] C. Meneveau and K. R. Sreenivasan. Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett., 59:1424–1427, 1987. [16] J. Neveu. Discrete-Parameter Martingales, volume 10 of North-Holland Mathematical Library. North Holland, Amsterdam, 1975. Translated from the French original by T.P. Speed. [17] M. Ossiander and E. Waymire. Statistical estimation for multiplicative cascades. To appear; available from {ossiand,waymire}@math.orst.edu, 1999. [18] S.I. Resnick. A Probability Path. Birkh¨ auser, Boston, 1998. [19] R. H. Riedi and J. Levy-Vehel. Tcp traffic is multifractal: A numerical study. Preprint, 1997. [20] Brent M. Troutman and Aldo V. Vecchia. Estimation of R´enyi exponents in random cascades. Bernoulli, 5(2):191–207, 1999.
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S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
School of Operations Research and Industrial Engineering and Department of Statistical Science, Cornell University, Ithaca, NY 14853 E-mail address: [email protected], [email protected]
AT&T Labs–Research, 180 Park Avenue, Florham Park, NJ 07932 E-mail address: [email protected], [email protected]
WAVELET ANALYSIS OF CONSERVATIVE CASCADES SIDNEY RESNICK, GENNADY SAMORODNITSKY, ANNA GILBERT, AND WALTER WILLINGER Abstract. A conservative cascade is an iterative process that fragments a given set into smaller and smaller pieces according to a rule which preserves the total mass of the initial set at each stage of the construction almost surely and not just in expectation. Motivated by the importance of conservative cascades in analyzing multifractal behavior of measured Internet traffic traces, we consider wavelet based statistical techniques for inference about the cascade generator , the random mechanism determining the re-distribution of the set’s mass at each iteration. We provide two estimators of the structure function, one asymptotically biased and one not, prove consistency and asymptotic normality in a range of values of the argument of the structure function less than a critical value. Simulation experiments illustrate the asymptotic properties of these estimators for values of the argument both below and above the critical value. Beyond the critical value, the estimators are shown to not be asymptotically consistent.
1. Introduction A multiplicative cascade is an iterative process that fragments a given set into smaller and smaller pieces according to some geometric rule and, at the same time, distributes the total mass of the given set according to another rule. The limiting object generated by such a procedure generally gives rise to a singular measure or multifractal – a mathematical construct that is able to capture the highly irregular and intermittent behavior associated with many naturally occurring phenomena, e.g., fully developed turbulence (see [10, 12, 4, 15] and references therein); spatial rainfall [6]; the movements of stock prices [14]; and Internet traffic dynamics [19, 3]. The generator of a cascade determines the re-distribution of the set’s total mass at every iteration; it can be deterministic or random. Cascade processes with the property that the generator preserves the total mass of the initial set at each stage of the construction almost surely and not just in expectation are called conservative cascades and are the main focus of this paper. Originally introduced by Mandelbrot [13] (also in the turbulence context), conservative cascades have recently been considered in [3] for use in describing the observed multifractal behavior of measured Internet traffic traces. In particular, Feldmann et al. [3] build on empirical evidence that measured Internet traffic is consistent with multifractal behavior by illustrating that “... data networks appear to act as conservative cascades!” They demonstrate that • multiplicative and measure-preserving structure becomes most apparent when analyzing measured Internet traces at a particular layer within the well-defined protocol hierarchy The visits of Sidney Resnick to AT&T Labs–Research were supported by AT&T Labs–Research and a National Science Foundation Grant from the Cooperative Research Program in the Mathematical Sciences. S. Resnick and G Samorodnitsky were also partially supported by NSF grant DMS-97-04982 and NSA grant MDA904-98-1-0041 at Cornell University. c
0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page
1
2
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
of today’s Internet Protocol (IP)-based networks, namely the Transport Control Protocol (TCP) layer and at the level of individual TCP connections, • this structure is recovered at the aggregate level (i.e., when considering the superposition process consisting of all IP packets generated by all active TCP connections) and causes aggregate Internet traffic to exhibit multifractal behavior. Well short of providing a physical explanation to the all-important networking question of “Why do packets within individual TCP connections conform to a conservative cascade?” the work in [3] is empirical in nature and relies on a number of heuristics for inferring multifractal behavior from traces of measured Internet traffic. However, to provide a more solid statistical basis for empirical studies of multifractal phenomena, progress in the area of statistical inference for multiplicatively generated multifractals is crucial. In this paper we contribute to the effort of providing rigorous techniques for multifractal analysis by investigating wavelet-based estimators for conservative cascades (i.e., for the class of multifractal processes generated by conservative cascades) and studying their large sample properties. In essence, the inference problem for conservative cascades consists of deducing from a single realization of the cascade process the distribution of the cascade generator that was presumably used to generate the sample or signal at hand. Intuitively, the generator’s distribution can be inferred from the degree of variability and intermittency exhibited locally in time by the signal under consideration. It can be expressed mathematically in terms of the local H¨older exponents which in turn characterize the singularity behavior of a signal locally in time. Moreover, since the local H¨older exponent at a point in time t0 describes the local scaling behavior of the signal as we look at smaller and smaller neighborhoods around t0 , a wavelet-based analysis that fully exploits the time- and scale-localization ability of wavelets proves convenient and is tailormade for our purpose. On the one hand, we exploit here the fact that the singularity behavior of a process can (under certain assumptions) be fully recovered by studying the singularity behavior in the wavelet domain; i.e., by investigating the (possibly) time-dependent scaling properties of the wavelet coefficients associated with the underlying process in the fine-time scale limit. On the other hand, using Haar wavelets, the discrete wavelet transform of a conservative cascade can be explicitly expressed in terms of the cascade’s generator (see for example [5]) and hence provides a promising setting for relating the local scaling behavior of the sample to the distribution of the underlying conservative cascade generator. In particular, we relate the distribution of the generator to an invariant of the cascade, namely the structure function or modified cumulant generating function (also known as Mandelbrot-Kahane-Peyri´ere (MKP) function [7]) and study the statistical properties (i.e., asymptotic consistency, asymptotic normality, confidence intervals) of two wavelet-based estimators of this function. Although the results in this paper have been largely motivated by our empirical investigations into the multifractal nature of measured Internet traffic [3, 5], we have clearly benefited from the recent random cascade work of Ossiander and Waymire [17]. Compared to the conservative cascades considered in this paper, random cascades are multiplicative processes with generators that preserve the total mass of the initial set only in expectation and not almost surely. This apparently minor difference ensures independence within and across the different stages of a random cascade construction but gives rise to subtle dependencies inherent in conservative cascades. Ossiander and Waymire [17] study the large sample asymptotics of estimators that are defined in the time-domain rather than in the wavelet-domain and allow for a rigorous statistical analysis of the scaling behavior exhibited by random cascades (for related work, see also [20]). While
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
3
the large-sample properties of the time domain-based estimators considered in [17] and of the wavelet-based estimators studied in this paper are very similar, their potential advantages, disadvantages and pitfalls when implementing and using them in practice require further studies. However, in combination, these different estimators provide a set of statistically rigorous techniques for multifractal analysis of highly irregular and intermittent data that are assumed to be generated by certain types of multiplicative processes or cascades. The rest of the paper is organized as follows. Sections 2–4 contain the basic facts about conservative cascades, their wavelet transforms, and some related quantities that are needed later in the paper. Section 5 discusses the critical constants and Section 6 is concerned with certain martingales and leads into Section 7 where subcritical asymptotics (that is, asymptotics for values of the argument below the critical value) and strong consistency of our two wavelet-based estimators is established. Asymptotic normality of the estimators is explained and illustrated with some simulated data in Section 8. We conclude in Section 9 with some supercritical asymptotics when the value of the arguement exceeds the critical value. The values of the estimators at large values of the argument of the structure function are uninformative and misleading, thus providing some practical guidance for properly interpreting the plots associated with the estimation procedure. 2. The Conservative Cascade. We now summarize the basic facts about the conservative cascade. Consider the binary tree. Nodes of the tree at depth l will be indicated by (j1 , . . . , jl ) ∈ {0, 1}l . Alternatively we consider successive subdivisions of the unit interval [0, 1]. After subdividing l times we have equal subintervals of length 2−l indicated by (2.1)
I(j1 , . . . , jl ) =
l l hX 1 jk X jk , + , 2k 2k 2l k=1
k=1
(j1 , . . . , jl ) ∈ {0, 1}l .
An infinite path through the tree is denoted by
j = (j1 , j2 , . . . ) ∈ {0, 1}∞
and the first l entries of j are denoted by
j|l = (j1 , . . . , jl ). We will sometimes write when convenient j|l, jl+1 = (j1 , . . . , jl , jl+1 ). The conservative cascade is a random measure on the Borel subsets of [0, 1] which may be constructed in the following manner. Suppose we are given a random variable W , called the d cascade generator , which has range [0, 1] and which is symmetric about 1/2 so that W = 1 − W. The symmetry implies that E(W ) = 1/2. We assume the random variable is not almost surely equal to 1/2. There is a family of identically distributed random variables {W (j|l), j ∈ {0, 1}∞ , l ≥ 1}
each of which is identically distributed as W . These random variables satisfy the conservative property (2.2)
W (j|l, 1) = 1 − W (j|l, 0).
4
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
Random variables associated with different depths of the tree are independent and random variables of the same depth which have different antecedents in the tree are likewise independent. Dependence of random variables having the same depth is expressed by (2.2). The conservative cascade is the random measure µ∞ defined by (2.3)
µ∞ (I(j|l)) =
l Y
W (j|i).
i=1
Note the conservative property entails that (2.4)
µ∞ (I(j|l, 0)) + µ∞ (I(j|l, 1)) = µ∞ (I(j|l)),
so that the weight of two offspring equals the weight of the parent. This implies X (2.5) µ∞ I(j|l) = 1. j|l
3. Wavelet Coefficients. We compute the wavelet transform Z 1 (3.1) ψ−l,n (x)µ∞ (dx), d−l,n =
n = 0, . . . , 2l − 1; l ≥ 1,
0
using the Haar wavelets
( 2l/2 , ψ−l,n (x) := −2l/2 ,
(3.2)
if if
2n ≤ x < 2n+1 , 2l+1 2l+1 2n+2 2n+1 ≤ x < 2l+1 . 2l+1
We have by examining where the Haar wavelet is constant that 2n + 1 2n + 2 2n 2n + 1 l/2 d−l,n = 2 µ∞ [ l+1 , l+1 ) − µ∞ [ l+1 , l+1 ) . 2 2 2 2 P Now suppose that lk=1 jk /2k = n/2l . Then we have from the definition (2.3) d−l,n =2l/2
l hY i=1
=2l/2
l Y i=1
W (j|i)W (j|l, 0) −
l Y i=1
i W (j|i)W (j|l, 1)
h i W (j|i) W (j|l, 0) − W (j|l, 1)
and using the conservative property (2.2), this is l/2
(3.3)
d−l,n =2
l Y i=1
for n = (3.4)
0, 1, . . . , 2l
h i W (j|i) 2W j|l, 0) − 1 ,
− 1. Sometimes where convenient, we will also write l h i Y W (j|i) 2W j|l, 0) − 1 . d−l,n = d(−l, j|l) = 2l/2 i=1
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
5
4. Notation Glossary. Before continuing the analysis, we collect some notation in one place for easy reference. We seek to estimate the distribution of the cascade generator W and this will be accomplished if we estimate c(q) := 2E(W q ),
(4.1)
q > 0,
or equivalently we could estimate the structure function (4.2)
τ (q) = 1 + log2 E(W q ) = log2 c(q).
The structure function will be estimated using estimators constructed from the process (4.3)
Z(q, l) =
l XY j|l i=1
W (j|i)q |2W (j|l, 0) − 1|q
and note from (3.3) that Z(q, l) =
(4.4)
1 2ql/2
l −1 2X
n=0
|d−l,n |q .
Our analysis rests on the process M (q, l), which we will show to be a martingale and which is defined as l 1 XY M (q, l) = (4.5) W (j|i)q , q > 0, l ≥ 1, c(q)l j|l i=1
and note the normalization makes E(M (q, l)) = 1. There are further constant functions needed: (4.6)
b(q) =E|2W − 1|q ,
(4.7)
a(q) =
(4.8)
ar (q) =
c(2q) E(W 2q ) = 2 , c2 (q) 2 E(W q )
21−r E(W rq ) c(rq) r . = cr (q) E(W q )
Note that a(q) = a2 (q). Finally we need three variances 1 (4.9) σ12 (q) := 2 Var(W q + (1 − W )q ), c 1 σ22 (q) := 2 Var(|2W − 1|q ). (4.10) b Wq (1 − W1 )q 1 1 (4.11) |2W2 − 1|q + |2W3 − 1|q − |2W1 − 1|q , σ32 (q) := 2 Var b c c where Wi , i = 1, 2, 3 are iid with the distribution of the cascade generator. It is convenient to define W = e−Y so that the Laplace transform of Y is (4.12)
φ(q) := Ee−qY = E(W q )
6
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
and ar (q) =
21−r φ(rq) . φr (q)
5. Critical Constants We now define the quantity (5.1)
q∗ := sup{q > 0 : a(q) < 1}
so that for q < q∗ we have a(q) < 1. It will turn out that when q < q∗ , the sequence {M (q, l), l ≥ 1} is an L2 -bounded and uniformly integrable martingale and this is the easiest case to analyze. It is always the case that q∗ ≥ 1, which follows from the fact that a(1) =
E(W 2 ) E(W 2 ) = 2E(W 2 ) = 2(E(W ))2 2( 12 )2
so that
1 1 1 1 1 a(1) = 2 Var(W ) + = 2E(W − )2 + ≤ 2 · |1 − |2 + = 1. 4 2 2 2 2 The Mandelbrot-Kahane-Peyri´ ere (MKP) Condition: Let W be the cascade generator and define Wq , q > 0, Xq = EW q so that EXq = 1. The MKP Condition is satisfied for q if (5.2)
E(Xq log2 Xq ) < 1
iff (5.3)
q E(W q log W ) − log E(W q ) < log 2 E(W q )
iff (5.4) Define
q(log φ)′ (q) − log φ < log 2 .
Λ∗ := {q : E(Xq log2 Xq ) < 1}. Then Λ∗ is an interval and we define the second critical constant (5.5)
q ∗ := sup Λ∗ .
Why is q ∗ considered a critical quantity? It turns out that the martingale {M (q, l), l ≥ 1} converges as l → ∞ to M (q, ∞) where ( 0, if q ≥ q ∗ , M (q, ∞) = something non-degenerate, if q < q ∗ . It turns out that the associated martingale is uninformative asymptotically when q > q ∗ . The two critical constants are related numerically by the inequality (5.6)
max(1, q ∗ /2) ≤ q∗ ≤ q ∗ .
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
7
To see why the inequality q∗ ≤ q ∗ in (5.6) is true, it suffices to show that if q > 0 satisfies a(q) < 1, then the MKP condition is satisfied for this q. However, if a(q) < 1, then log 2 > log φ(2q) − 2 log φ(q) = log φ(2q) − log φ(q) − log φ(q) Z 2q (log φ)′ (s)ds − log φ(q) = q
and since log φ is convex, (log φ)′ is increasing and the forgoing is bounded below by Z 2q ds − log φ(q) ≥(log φ)′ (q) q
=q(log φ)′ (q) − log φ(q).
We conclude
log 2 > q(log φ)′ (q) − log φ(q) which is equivalent to the MKP condition holding by (5.4). On the other hand, suppose that q∗ < ∞. Since a(q∗ ) = 1, we have in the same way as above log 2 = log φ(2q∗ ) − 2 log φ(q∗ ) = 2 log φ(2q∗ ) − log φ(q∗ ) − log φ(2q∗ ) Z 2q∗ (log φ)′ (s)ds − log φ(2q∗ ) =2 q∗
′
≤2(log φ) (2q∗ )
Z
2q∗
q∗
ds − log φ(2q∗ )
=2q(log φ)′ (2q∗ ) − log φ(2q∗ ).
Therefore, the MKP condition does not hold for 2q∗ , and so q ∗ ≤ 2q∗ .
Example 1. Suppose W is uniformly distributed on [0, 1]. In this case E(W q ) = 1/(1 + q) and so 1 q2 a(q) = 1+ 2 2q + 1 and √ q∗ = 1 + 2 ≈ 2.4. Likewise, q ∗ satisfies the equation q log(1 + q) − = log 2 1+q and so q ∗ ≈ 3.311. Example 2. Suppose more generally that W has the beta distribution with mean 1/2 (i.e., the shape parameters α and β are equal). Then E(W q ) =
(5.7) and q∗ satisfies
Γ(2α)Γ(α + q) Γ(α)Γ(2α + q)
√ 4−α πΓ(α + 2q∗ )Γ(2α + q∗ )2 − Γ(α + 1/2)Γ(2α + 2q∗ )Γ(α + q∗ )2 = 0.
8
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
Example 3. Suppose W has the two point distribution concentrating mass 1/2 at ±p for some 0 ≤ p < 1/2. Then 1 (5.8) E(W q ) = (pq + (1 − p)q ) 2 and 2pq (1 − p)q a(q) = 1 − q . (p + (1 − p)q )2 Note in this case that a(q) ↑ 1 as q ↑ ∞ so q∗ = q ∗ = ∞. Example 4. If W does not have a two point distribution but nevertheless has an atom of size p1 < 1/2 at 1 (and hence by symmetry there is an atom of the same size at 0) we have q ∗ < ∞ (and, hence, also q∗ < ∞). To see this we express the condition (5.3), when q > 1, in the equivalent form E(W q log2 W q ) > 1. (5.9) E(W q ) log2 2E(W q ) Note that if W does not have a two point distribution, then for q > 1, we have P [W q < W ] > 0 and E(W q ) < E(W ) = 1/2, so log2 (2E(wq )) < 0, which explains the sign reversal in (5.9) compared with (5.3). By the dominated convergence theorem the numerator in (5.9) converges to 0 as q → ∞, while the denominator converges to P [W = 1] log2 2P [W = 1] 6= 0. Hence, (5.9) fails for large q. Based on the experience provided in Examples 3 and 4, it natural to wonder how common it can be that q ∗ = ∞. This is discussed in the next proposition.
Proposition 5.1. Unless W has a two point distribution, it must be the case that q ∗ < ∞.
Proof. Because of Example 3, we may assume that W does not have atoms at 0 and 1. Let p ∈ [0, 1/2) be the leftmost point of the support of the distribution of W . Then 1 − p is the rightmost point of the support of the distribution of W . For 0 < ρ < 1, we have θ(ρ) := P [W ≥ ρ(1 − p)] > 0.
Since the distribution of W is not a two point distribution, limρ→1 θ(ρ) < 12 . Thus, we can find and fix a value of 0 < ρ < 1 such that θ(ρ) < 1/2,
0 < ρ < 1.
For this value of ρ, it is convenient to set δ(ρ) := δ = ρ(1 − p).
Apply Jensen’s inequality with the convex function g(x) = x log x, x > 0, to get q q Wq W q W W · 1[W ≥δ] =θ(ρ)E log2 log2 E W ≥ δ c(q) c(q) c(q) c(q) q q W W ≥θ(ρ)E W ≥ δ log2 E W ≥ δ c(q) c(q) Wq W q =E 1 (5.10) W ≥ δ . · log E 2 2E(W q ) [W ≥δ] 2E(W q )
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
9
Also we have Wq 1 lim E q→∞ c(q) [W ≥δ]
(5.11)
1 = . 2
To verify (5.11), note that W q) Wq Wq 1 =E =E 1[W ≥δ] + E 1[W 0 log 2 2 2EW q E(W q ) 2 2 ∗ and so the MKP condition (5.3) fails for all large q. Therefore, q < ∞. E
10
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
5.1. Properties of the function ar (q). We need the following properties of the function ar (q). Proposition 5.2. (i) For any fixed r > 1, the function ar (q) (and therefore a(q)) is strictly increasing in q > 0. (ii) For any fixed q > 0, the function log ar (q) is strictly convex in the region r > 1. (iii) If q satisfies the MKP condition, then d log ar (q) < 0, dr r=1 and there exists r0 ∈ (1, 2) such that
(5.15)
ar0 (q) < a1 (q) = 1.
(iv) If the MKP condition fails for q and the inequalities in (5.2), (5.3) or (5.4) are reversed to become strictly greater than, we have d (5.16) log ar (q) > 0, dr r=1 and there exists 0 < r1 < 1 and ar1 (q) < a1 (q) = 1. Proof. (i) Recall the definition of φ from (4.12). For fixed r > 1, if we differentiate with respect to q, we get φ(rq) ′ φr (q)rφ′ (rq) − φ(rq)rφr−1 (q)φ′ (q) = . φr (q) φ2r (q) This is positive iff φ(q)φ′ (rq) > φ(rq)φ′ (q) or φ′ (q) φ′ (rq) > . φ(rq) φ(q) Since r > 1, it suffices to show φ′ /φ is strictly increasing which is true if its derivative is strictly positive. The derivative is φ(q)φ′′ (q) − (φ′ (q))2 φ2 (q) and this is strictly positive iff φ(q)φ′′ (q) > (φ′ (q))2 ,
(5.17) that is iff
2 E(e−qY )E(Y 2 e−qY ) > E(Y e−q/2Y · e−q/2Y )
which follows from the Cauchy–Schwartz inequality. (ii) Fix q > 0 and check that
q 2 ′′ d2 (log a (q)) = φ (rq)φ(rq) − (φ′ (rq))2 r 2 2 dr φ (rq)
which is positive by (5.17). (iii) For fixed q > 0,
d log ar (q) = q(log φ)′ (qr) − log 2 − log φ(q), dr
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
so that
d d dr ar (q) log ar (q) = dr ar (q) r=1
r=1
11
= q(log φ)′ (q) − log φ(q) − log 2 < 0.
Since a1 (q) = 1, we have
d ar (q) < 0, dr r=1 Hence there exists r0 ∈ (1, 2) such that
a1 (q) = 1.
ar0 (q) < a1 (q) = 1.
(iv) If d a (q) d r = dr log ar (q) dr ar (q) r=1
r=1
= q(log φ)′ (q) − log φ(q) − log 2 > 0,
then since log a1 (q) = 0, there exists r1 < 1 such that log ar1 (q) < 0 or ar1 (q) < 1.
6. The Associated Martingale. In this section we study the properties of the process {M (q, l), l ≥ 1} defined in (4.5) for each fixed q > 0. We define the increasing family of σ-fields Fl := σ{W (j|l), j|l ∈ {0, 1}l }
generated by the weights up to and including depth l. Proposition 6.1. For each q > 0, the family
{(M (q, l), Fl ), l ≥ 1}
is a non-negative martingale with constant mean 1 such that M (q, l) converges almost surely to a limiting random variable M (q, ∞): a.s.
M (q, l) → M (q, ∞),
If the MKP condition fails for q, then
E(M (q, ∞) ≤ 1.
P [M (q, ∞) = 0] = 1,
and if q satisfies the MKP condition, then E(M (q, ∞)) = 1 so that P [M (q, ∞) > 0] = 1.
Proof. The martingale property is easily established: X Ql W q (j|i) W q (j|l, jl+1 ) i=1 |Fl E(M (q, l + 1)|Fl ) = E cl c j|l,jl+1
l XY W q (j|i) X q = E W (j|l, j )/c l+1 cl j|l i=1
jl+1
q
=M (q, l)2 · E(W )/c = M (q, l).
12
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
By the martingale convergence theorem (eg, [18], [16]) a non-negative martingale always converges almost surely. The last statements follow by the methods of Kahane and Peyri´ere ([8]). See also Propositions 6.2 and 6.3. Example 5. Recall the example of the two point distribution of Example 3 of Section 4. In this case we have M (q, l) = 1 for all q > 0 and l ≥ 1. For verifying this, the key observation is that W q + (1 − W )q = pq + (1 − p)q .
(6.1)
Recall (5.8) and then observe for l > 1 l l−1 XY XY W q (j|i) W q (j|i) X W q (j|l − 1, jl ) M (q, l) = = c c c j|l i=1
=
X
l−1 Y
j|l−1 i=1
jl
j|l−1 i=1
W q (j|i) c
h
i W q (j|l − 1, 0) + W q (j|l − 1, 1) /c
and since W (j|l − 1, 1) = 1 − W (j|l − 1, 0) we apply (6.1) to get =
l−1 XY W q (j|i) h c
j|l−1 i=1
=
l−1 XY W q (j|i) c
j|l−1 i=1
i pq + (1 − p)q /c
= M (q, l − 1).
One can easily see that M (q, 1) = 1 and the assertion is shown. Define M (q, 0) = 1 and let the martingale differences be d(q, l) := M (q, l) − M (q, l − 1),
l ≥ 1.
For l > 1 we have from the definition of M (q, l) that l−1 i XY W q (j|i) h W q (j|l − 1, 0) + (1 − W (j|l − 1, 0)q d(q, l) = −1 c c j|l−1 i=1
(6.2)
=
l−1 XY W q (j|i) h
j|l−1 i=1
c
i ξ(j|l) .
We now easily see E(d(q, l)|Fl−1 ) = 0. For the conditional variance, note E(ξ(j|l)) = 0 and recall the notation from (4.9) 1 σ12 (q) = Var(ξ(j|l) = 2 Var(W q + (1 − W )q ). c So the conditional variance of d(q, l) is ! l−1 2 X Y q (j|i) W ξ(j|l) |Fl−1 E(d2 (q, l)|Fl−1 ) =E c j|l−1 i=1
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
=
l−1 l−1 X Y W q (j|i) Y W q (p|i) c
j|l−1 i=1 p|l−1
c
i=1
13
E(ξ(j|l)ξ(p|l)).
Since ξ(p|l)) ⊥ ξ(j|l)) if p|l 6= j|l we have 2
E(d (q, l)|Fl−1 ) =
l−1 X Y W q (j|i) 2 c
j|l−1 i=1
=
l−1 XY W 2q (j|i) c(2q)
j|l−1 i=1
σ12 (q)
al−1 (q)σ12 (q)
=M (2q, l − 1)al−1 (q)σ12 (q). Thus the conditional variance of M (q, l) is (6.3)
l X
E(d2 (q, i)|Fi−1 ) =
i=1
i=1
Furthermore, and thus (6.4)
l X
M (2q, i − 1)ai−1 (q)σ12 (q).
E(d2 (q, l)) = E(E(d2 (q, l)|Fl−1 )) = EM (2q, l − 1)al−1 (q)σ12 (q) = al−1 (q)σ12 (q) Var(M (q, l)) =
l X
E(d2 (q, i)) = σ12 (q)
ai−1 (q).
i=1
i=1
This leads to the following facts.
l X
Proposition 6.2. If q < q∗ so that a(q) < 1, the martingale {(M (q, l), Fl ), l ≥ 0} is L2 -bounded and hence uniformly integrable. It follows that (6.5)
E(M (q, ∞) = 1,
M (q, l) = E(M (q, ∞)|Fl ) ,
and M (q, l) → M (q, ∞) almost surely and in L2 . Moreover, if q∗ ≤ q < q ∗ , then the martingale {(M (q, l), Fl ), l ≥ 0} is Lp -bounded for some 1 < p < 2 and, hence, still uniformly integrable, (6.5) still holds and M (q, l) → M (q, ∞) almost surely and in Lp .
Remark. The proof will show that when q∗ ≤ q < q ∗ , we may take p = r0 , where r0 is given in Proposition 5.2 (iii). See (5.15). Proof. Suppose first that q < q∗ . We have from (6.4) that sup E(M (q, l) − 1)2 = sup Var(M (q, l)) l≥0
l≥0
= lim ↑ l→∞
=
∞ X i=1
l X
E(d2 (q, i)
i=1
ai−1 (q)σ12 (q) < ∞.
14
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
The rest follows from standard martingale theory (eg, [16, page 68], [18]). Now let q < q ∗ and we consider uniform integrability without L2 boundedness. Suppose 1 < p ≤ 2 and for two paths j 1 and j 2 denote by mj1 ,j2 the largest i ≤ l such that j 1 |i = j 2 |i. We have p/2 X X Y l 1 q q W (j 1 |i)W (j 2 |i) E E(M (q, l)) = c(q)lp p
j1 |l j 2 |l i=1
X l 1 X ≤ E c(q)lp k=0
l Y i=1
j 1 |l j 2 |l
p/2 W (j 1 |i)W (j 2 |i) q
q
mj 1 ,j 2 =k
X Y k l 1 X W 2q (j|i) · E ≤ lp c (q) j|k i=1
k=0
X
(1) (1) jk+1 ,...,jl (2) (2) jk+1 ,...,jl (1) (2) jk+1 6=jk+1
l Y
W
q
i=k+1
p/2
(1) (1) (2) (2) (j1 , . . . , jk , jk+1 , . . . , jl |i)W q (j1 , . . . , jk , jk+1 , . . . , jl |i)
l
≤
1 X X pq k E W · clp (q) k=0 j|k
E
≤
X
k=0
cp (q)
E
W
q
c(l−k)p (q)
·
(1) (1) jk+1 ,...,jl (2) (2) jk+1 ,...,jl (2) (1) jk+1 6=jk+1
l X c(pq) k
(1)
X
l Y
i=k+1
1
(1)
jk+1 ,...,jl
i=k+1
(2) (2) jk+1 ,...,jl (1) (2) jk+1 6=jk+1
=
l X k=0
=
k
ap (q)
l Y
1 c(l−k)p (q)
E(W (1 − W ))q
p/2
(2) (2) (1) (1) (j1 , . . . , jk , jk+1 , . . . , jl |i)W q (j1 , . . . , jk , jk+1 , . . . , jl |i)
p/2 (2) (2) (1) (1) W q (j1 , . . . , jk , jk+1 , . . . , jl |i)W q (j1 , . . . , jk , jk+1 , . . . , jl |i)
q
q 2(l−k−1) 2(l−k−1)
E(W (1 − W )) (EW )
p/2 2p/2 c(q)
X l k=0
ap (q)k .
2
p/2 ·2
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
15
Here a product over the empty set is equal to 1. By Proposition 5.2 (iii) there is a p ∈ (1, 2) such that ap (q) < 1. For this p the martingale {(M (q, l), Fl ), l ≥ 0} is Lp -bounded, and the rest follows, once again, from standard martingale theory. 6.1. The distribution of M (q, ∞). The distribution of M (q, ∞) satisfies a simple recursion which can be used to derive additional information. Proposition 6.3. Suppose {M (q, ∞), M1 (q, ∞), M2 (q, ∞)} are iid with the same distribution as M (q, ∞), the martingale limit. Let W have the distribution of the cascade generator and suppose W and {M (q, ∞), M1 (q, ∞), M2 (q, ∞)} are independent. Then d
M (q, ∞) = W q
(6.6)
M2 (q, ∞) M1 (q, ∞) + (1 − W )q c(q) c(q)
and for any q > 0, (6.7)
P [M (q, ∞) = 0] = 0 or 1,
so that E(M (q, ∞)) = 1 implies P [M (q, ∞) = 0] = 0. Proof. We write M (q, ∞) = lim
l→∞
l XY W q (j|i)
cl
j|l i=1
l l X Y X Y W q (1, j2 , . . . , jl ) W q (0, j2 , . . . , jl ) + = lim l→∞ cl cl j2 ,...,jl i=1
j2 ,...,jl i=1
X = lim W q (0) l→∞
l Y
j2 ,...,jl i=2
l X Y W q (1, j2 , . . . , jl ) W q (0, j2 , . . . , jl ) q + (1 − W (0)) cl cl j2 ,...,jl i=2
M1 (q, ∞) M2 (q, ∞) + (1 − W (0))q . c c Now we verify (6.7). Define d
=W q (0)
p0 =P [M (q, ∞) = 0]
pW (0) =P [W = 0] = P [W = 1]. Then since c(q) 6= 0
p0 =P [M (q, ∞) = 0] = P [W q M1 (q, ∞) + (1 − W )q M2 (q, ∞) = 0] =P [′′ , W = 0] + P [′′ , W = 1] + P [′′ , 0 < W < 1].
¿From this we conclude so that
p0 = 2pW (0)p0 + (1 − 2pW (0))p20
p0 (1 − 2pW (0)) = p20 (1 − 2pW (0)). If 0 < pW (0) < 1/2, then p0 = p20 and p0 = 0 or 1. If pW (0) = 1/2, then P [W = 0] = P [W = 1] = 1/2 and W has a two point distribution and hence from Example 5 we know M (q, l) = 1 which implies M (q, ∞) = 1.
16
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
7. Estimation: Subcritical Consistency We propose two estimators of the structure function which depend on scaled summed powers of the wavelet coefficients {Z(q, l), l ≥ 1}. These are P l −1 log2 2n=0 |d−l,n |q − ql/2 log2 Z(q, l) (7.1) = τˆ1 (q) = τˆ1 (q, l) = l l ! P2l+1 −1 q |d | Z(q, l + 1) −(l+1),n n=0 τˆ2 (q) = τˆ2 (q, l) = log2 (7.2) . = log2 P2l −1 q/2 Z(q, l) 2 |d−l,n |q n=0
Analysis depends on showing that scaled versions of Z(q, l) are well-approximated by the martingale and this is discussed next. Recall notational definitions (4.1), (4.2), (4.3), (4.6). Proposition 7.1. For q > 0, Z(q, l) P − M (q, l) → 0. l cb If q = 6 q ∗ , the convergence is almost sure and if q < q∗ , the convergence is in L2 . Thus Z(q, l) → M (q, ∞) cl b in the appropriate sense, depending on the case. (7.3)
Proof. Begin by writing l i XY Z(q, l) W q (j|i) h |2W (j|l, 0) − 1|q − M (q, l) = − 1 cl b c b j|l i=1
(7.4)
=
l XY W q (j|i)
c
j|l i=1
ξ(j|l, 0)
where ξ(j|l, 0) ⊥ ξ(p|l, 0) if j|l 6= p|l. Also Eξ(j|l, 0) = 0 and recall the notation from (4.10)
1 Var(|2W − 1|q ). b2 If q < q∗ , so a(q) < 1, then similar to the calculations leading to (6.3) and (6.4) we find ! l l 2 X X Z(q, l) Y W q (j|i) Y W q (p|i) − M (q, l) = ξ(j|l, 0)ξ(p|l, 0) E E cl b c c σ22 (q) := Eξ 2 (j|l, 0) =
j|l p|l
=σ22 (q)
i=1
(2EW 2q )l c2l (q)
i=1
= σ22 (q)al (q)
→0 as l → ∞ since a(q) < 1. This shows the L2 –convergence. For q > 0, the same method shows Z(q, l) 2 |Fl =σ22 (q)M (2q, l)al (q) − M (q, l) E cl b
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
=σ22 (q)
(7.5)
l XY W 2q (j|i) j|l i=1
c2 (q)
17
=: σ22 (q)V (q, l),
and we need to show V (q, l) → 0, almost surely as l → ∞. If the MKP condition fails, then M (q, l) → 0 as l → ∞ and 2 V (q, l) ≤ M (q, l) → 0. If the MKP condition holds, then from Proposition 5.2 (iii), there exists r0 ∈ (1, 2) such that ar0 (q) < 1, and for p = r0 /2 ∈ (1/2, 1) we have by the triangle inequality (7.6)
0 ≤ V (q, l)p ≤
l XY W 2pq (j|i) j|l i=1
c2p (q)
a.s.
= (ar0 (q))l M (r0 q, l) → 0,
as l → ∞, since M (r0 q, ∞) < ∞ almost surely. So in all cases V (q, l) → 0. For any δ > 0, ǫ > 0 we have Z(q, l) Z(q, l) − M (q, l) > ǫ|Fl ] =P [ l − M (q, l) > ǫ|Fl ]1[V (q,l)σ22 (q)>δ] P [ l cb cb Z(q, l) + P [ l − M (q, l) > ǫ|Fl ]1[V (q,l)σ22 (q)≤δ] cb ! Z(q, l) 2 ≤1[V (q,l)σ22 (q)>δ] + ǫ−2 E − M (q, l) |Fl 1[V (q,l)σ22 (q)≤δ] cl b =1[V (q,l)σ22 (q)>δ] + ǫ−2 V (q, l)σ22 (q)1[V (q,l)σ22 (q)≤δ] ≤1[V (q,l)σ22 (q)>δ] +
δ2 . ǫ2
a.s.
Take expectations and use V (q, l) → 0 and the arbitrariness of δ to conclude Z(q, l) P − M (q, l) → 0, l cb
as l → ∞. For almost sure convergence, when q < q ∗ , we get from (7.6) that l V (q, l) ≤ ar0 (q)1/p M 1/p (r0 q, l) P and so l V (q, l) < ∞ almost surely. Thus, for any ǫ > 0, ! Z(q, l) 2 X X X Z(q, l) E −M (q, l) |F = (const) V (q, l) < ∞, P [| l −M (q, l)| > ǫ|Fl ] ≤ ǫ−2 l cb cl b l
l
and by a generalization of the Borel-Cantelli lemma ([16, page 152]) we have
Z(q, l) a.s. − M (q, l) → 0. cl b For q > q ∗ , we prove almost sure convergence from Proposition 5.2 (iv) in a similar way. We use this comparison result Proposition 7.1 to get consistent estimators of the structure function τ (q) in the subcritical case, by which we mean the case where the MKP condition holds.
18
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
Proposition 7.2. Define τˆi (q) for i = 1, 2 by (7.1) and (7.2). Provided q < q ∗ , so that the MKP condition holds, both estimators are almost surely consistent for τ (q): a.s.
τˆi (q) → τ (q),
i = 1, 2,
as l → ∞. Proof. In (7.3), take logarithms to the base 2 to get (7.7)
log2 Z(q, l) − l log2 c(q) − log2 b → log2 M (q, ∞),
almost surely as l → ∞. Divide through by l to get consistency of τˆ1 (q). To get the consistency of τˆ2 (q), note from (7.7) that log2 Z(q, l + 1) − log2 Z(q, l) − (l + 1 − l)τ (q) → 0 almost surely which proves consistency of τˆ2 (q). 8. Subcritical Asymptotic Normality of Estimators. In this section we discuss second order properties of the estimators τˆi (q), i = 1, 2, defined in (7.1) and (7.2). The asymptotic normality for τˆ1 (q) requires a bias term which cannot be eliminated. This drawback, is eliminated by using τˆ2 (q), whose definition in terms of differencing removes the bias term. However, take note of the suggestive remarks at the end of this Section 8 about mean squared error. For this section it is convenient to write E Fl and P Fl for the conditional expectation and conditional probability with respect to the σ-field Fl . 8.1. Asymptotic Normality of τˆ1 (q). Begin by writing (8.1) (8.2)
l i XY W q (j|i) h |2W (j|l, 0) − 1|q Z(q, l) − M (q, l) = − 1 cl b c b j|l i=1 X =: Z(j|l) j|l
where E Fl (Z(j|l)) =0 Y l W 2q (j|i) l 2 Fl E (Z(j|l)) = a (q)σ22 (q), c(2q) i=1
and recall σ22 (q) is defined in (4.10). Therefore, X (8.3) E Fl (Z(j|l))2 = M (2q, l)al (q)σ22 (q). j|l
Our strategy for the central limit theorem is to regard Z(q,l) −M (q, l) as a sum of random variables cl b which are conditionally independent given Fl and then apply the Liapunov condition ([18]) for asymptotic normality in a triangular array.
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
Proposition 8.1. If 2q < q ∗ , then as l → ∞ # " Z(q,l) − M (q, l) l (8.4) ≤ x → P [N (0, 1) ≤ x] P Fl p c b M (2q, l)al (q)σ22 (q)
19
a.s.
where N (0, 1) is a standard normal random variable with mean 0 and variance 1. Taking expectations in (8.4) yields # " Z(q,l) − M (q, l) lb c (8.5) ≤ x → P [N (0, 1) ≤ x]. P p M (2q, l)al (q)σ22 (q) Proof. By Proposition 5.2 (iii) there is δ > 0 such that both 2q + δ < q ∗ and
(8.6)
a1+δ/2 (2q) < 1.
Asymptotic normality in (8.4) will be shown if we establish the Liapunov condition P 2+δ Fl j|l E |Z(j|l)| (8.7) →0 a.s., (M (2q, l)al (q))(2+δ)/2 where the denominator comes from (8.3). The numerator in the left side of (8.7) is bounded above by 2+δ l 2+δ |2W (j|l, 0) − 1|q q (j|i) X Y W − 1 E Fl c b j|l
i=1
=c1
l XY W q(2+δ) (j|i) cl ((2 + δ)q) j|l
i=1
c((2 + δ)q)
cl(2+δ) (q)
=c1 M ((2 + δ)q, l)(a2+δ (q))l , where
|2W (j|l, 0) − 1|q 2+δ c1 = E − 1 . b So the ratio in (8.7), apart from constants, is bounded by
M ((2 + δ)q, ∞)(a2+δ (q))l M ((2 + δ)q, l)(a2+δ (q))l ∼ . M (2q, l)1+δ/2 (a2 (q))(1+δ/2)l M (2q, ∞)1+δ/2 (a2 (q))(1+δ/2)l
Note that the two random variables M ((2 + δ)q, ∞) and M (2q, ∞) are non zero with probability 1 by Proposition 6.1. Check that a2+δ (q) = a1+δ/2 (2q) < 1. (a2 (q))1+δ/2 So the Liapunov ratio is asymptotic to a finite nonzero random variable times (a1+δ/2 (2q))l where a1+δ/2 (2q) < 1 and the result is proven. Remark 8.1. In the denominator of (8.5) we may replace M (2q, l) by its limit M (2q, ∞). This follows since almost surely 0 < M (2q, ∞) < ∞ for 2q < q ∗ and thus s ! Z(q,l) − M (q, l) M (2q, l) l p cb , ⇒ (N (0, 1), 1) M (2q, ∞) M (2q, l)al (q)σ22 (q)
20
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
by [2]. The desired result is obtained by multiplying components. Remark 8.2. Set Z(q,l) − M (q, l) l . Nl := p c b M (2q, l)al (q)σ22 (q)
(8.8) Then in R2 , as l → ∞
(Nl , Nl+1 ) ⇒ N1 (0, 1), N2 (0, 1) , where Ni (0, 1), i = 1, 2 are iid standard normal random variables. To see this, write for any x, y ∈ R
P [Nl ≤ x, Nl+1 ≤ y] =EP Fl+1 [Nl ≤ x, Nl+1 ≤ y] =E1[Nl ≤x] P Fl+1 [Nl+1 ≤ y].
By Proposition 8.1,
P Fl+1 [Nl+1 ≤ y] = Φ(y) + ǫl (y)
a.s.
L1
where Φ(y) is the standard normal cdf and where ǫl (y) → 0 and |ǫl (y)| ≤ 2. So P [Nl ≤ x, Nl+1 ≤ y] =E1[Nl ≤x] Φ(y) + ǫl (y) =E1[Nl ≤x] Φ(y) + o(1)
from the dominated convergence theorem, and hence we get convergence to →Φ(x)Φ(y). We now describe how this central limit behavior transfers to τˆ1 (q). Corollary 8.1. Under the assumptions in force in Proposition 8.1, we have τˆ1 (q) − τ (q) − l−1 log2 bM (q, l) √ (8.9) ⇒ N (0, 1). 2 l M (2q,∞)a (q)σ2 (q) l log 2·M (q,l)
Remark. The bias term l−1 log2 bM (q, l) cannot be neglected. Proof. For brevity, write
(8.10)
d(q) := M (2q, ∞)al (q)σ22 (q),
and using the notation of (8.8) we have
Since
p Z(q, l) = cl b Nl d(q) + M (q, l) .
1 log2 Z(q, l), l p lˆ τ1 (q) = l log2 c + log2 b + log2 Nl d(q) + M (q, l) τˆ1 (q) =
we have and thus
p Nl d(q) . l τˆ1 (q) − τ (q) = log2 bM (q, l) + log2 1 + M (q, l)
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
21
∗ Since p by (5.6) and assumption 2q < q we have q < q∗ , we know that d(q) → 0. Therefore, Nl d(q)/M (q, l) → 0, and the desired result follows by using the relation log(1 + x) ∼ x for x ↓ 0.
The bias term in (8.9) is an unpleasant feature and thus we consider how to remove it by differencing. 8.2. Asymptotic Normality of τˆ2 (q). We now consider the asymptotic normality of τˆ2 (q). It is possible to proceed from Proposition 8.1 but it turns out to be simpler to proceed with a direct proof. Proposition 8.2. Suppose 2q < q ∗ . Then √
(8.11)
τˆ2 (q) − τ (q)
M (2q,∞)al (q)σ32 (q) log 2·M (q,∞)
⇒ N (0, 1),
where σ32 (q) is defined in (4.11). Proof. Begin by observing that Z(q, l + 1) Z(q, l) − cl+1 b cl b " q l X Y W q (j|i) W q (j|l, 1) |2W (j|l, 1, 0) − 1|q W (j|l, 0) |2W (j|l, 0, 0) − 1|q = + c c b c b j|l i=1 # |2W (j|l, 0) − 1|q − b l X Y W q (j|i) H(j|l) =: c j|l
i=1
where we have set (8.12) H(j|l) =
W q (j|l, 1) |2W (j|l, 1, 0) − 1|q |2W (j|l, 0) − 1|q W q (j|l, 0) |2W (j|l, 0, 0) − 1|q + − . c b c b b
Check that 1 1 E Fl H(j|l) = + − 1 = 0 2 2 and E Fl H 2 (j|l) =σ32 (q). It follows that conditionally on Fl we may treat Z(q, l + 1) Z(q, l) − cl+1 b cl b
22
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
as a sum of iid random variables with (conditional) variance 2 l Z(q, l) 2 X Fl Y W q (j|i) Fl Z(q, l + 1) H(j|l) E = E − cl+1 b cl b c(q) i=1
j|l
=
l XY W 2q (j|i) l a (q)EH 2 (j|l) c(2q) j|l
i=1
=M (2q, l)al (q)σ32 (q).
As in the proof of Proposition 8.1 we may check the Liapunov condition and conclude Z(q,l) cl b M (2q, ∞)al (q)σ32 (q) Z(q,l+1) cl+1 b
p
or equivalently
−
⇒ N (0, 1),
Z(q,l) − 1 c−1 Z(q,l+1) Z(q,l) cl b p ⇒ N (0, 1), M (2q, ∞)al (q)σ32 (q)
and since Z(q, l)/(cl b) → M (q, ∞) we have c−1 Z(q,l+1) − 1 M (q, ∞) Z(q,l) p (8.13) ⇒ N (0, 1). M (2q, ∞)al (q)σ32 (q) Since
c−1
Z(q, l + 1) P − 1 → 0, Z(q, l)
it follows that Z(q, l + 1) τˆ2 (q) − τ (q) = log2 c−1 Z(q, l) log 1 + c−1 Z(q,l+1) − 1 Z(q,l) = log 2 ∼
c−1 Z(q,l+1) Z(q,l) − 1 log 2
in probability. Combine this with (8.13) to complete the proof. For statistical purposes, the result (8.11) contains unobservables so as in [20, 17], consideration needs to be given to replacing quantities which are not observed by observable estimators. We assume that the random measure µ∞ is observed, or equivalently that the wavelet coefficients {d−l,n } are known. This means we have the quantities {Z(q, l)}. We define the following useful observable quantity " l XY W q (j|l, 1)|2W (j|l, 1, 0) − 1|q W q (j|l, 0)|2W (j|l, 0, 0) − 1|q + W 2q (j|i) D 2 (q, l) = Z(q, l + 1) Z(q, l + 1) j|l i=1
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
|2W (j|l, 0) − 1|q − Z(q, l)
(8.14) =:
l XY
23
#2
W 2q (j|i)V 2 (j|l)
j|l i=1
=:
l XY j|l i=1
"
C A+B W 2q (j|i) − Z(q, l + 1) Z(q, l)
#2
.
Note that in this notation, A+B C − , cb b 2 where H(j|l) is defined in (8.12). Recall also that EH (j|l) = σ32 (q). In terms of the wavelet coefficients we have (8.15) #2 " q 2−q(l+1)/2 q 2−ql/2 X |d(−l, (j|l, 0))|q 2−q(l+1)/2 |d(−l, (j|l, 1))| |d(−l, (j|l)| , + − D 2 (q, l) = Z(q, l + 1) Z(q, l + 1) Z(q, l) H(j|l) =
j|l
showing that D 2 (q, l) is an observable statistic. Corollary 8.2. Suppose 2q < q ∗ . Then τˆ2 (q) − τ (q) ⇒ N (0, 1) D(q, l)/ log 2
(8.16) as l → ∞.
Proof. Because of (8.11), it suffices to show D 2 (q, l) P → 1, M (2q, ∞)al (q)σ32 (q)/M 2 (q, ∞)
as l → ∞. This is equivalent to showing
Z 2 (q,l) D 2 (q, l) P c2l (q)b2 (q) → M (2q, l)al (q)σ32 (q)
1.
After some simple algebra, this ratio is the same as ! #2 P Ql W 2q (j|i) " C Z(q, l)/cl b A+B j|l i=1 c(2q) − . bc Z(q, l + 1)/cl+1 b b M (2q, l)σ32 (q) Since M (2q, l) → M (2q, ∞), it suffices to show that the numerator converges in probability to M (2q, ∞)σ32 (q). Due to (7.3), we write the numerator as ! l A + B 2 A + B C XY W 2q (j|i) h A + B C i2 + op (1)2 + op (1)2 − − c(2q) bc b bc b bc j|l i=1
=I + II + III.
24
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
q = 0.25, shape = 1 0.10
q = 0.75, shape = 1
0.05
0.08
-0.05
0.06 0.04
10 12 14 16
tau2 0.0
tau1 0.10
0.12
0.14
10 12 14 16
-2
-1 0 1 Quantiles of standard normal
2
-2
-1 0 1 Quantiles of standard normal
2
Figure 1. Normal QQ-plots of τˆ1 (q) − τ (q) (left) and τˆ2 (q) − τ (q) (right). As in Theorem 3.5 of [17], A + B
C 2 = M (2q, ∞)σ32 (q), bc b as desired. The terms I and II can readily be shown to go to 0. I → M (2q, ∞)E
−
Figure 1 shows normal QQ plots of τˆi (q) − τ (q), i = 1, 2, from simulated cascade data with beta distributed cascade generator with shape parameter 1 (this makes the distribution uniform). The left plot is for q = 0.75 and the right is for q = 0.25. Each plot presents 4 graphs as the depth l increases to 16. Note the better agreement of τˆ2 (q) to normality compared with τˆ1 (q). Concluding remarks on mean squared error: Examining Corollary 8.1 and Proposition 8.2 yields that in the region 2q < q ∗ , the conditional mean squared error of τˆ1 (q) is of the form (1)
(2)
Op (al (q))2 Op (1) + l2 l2 (3)
while that of τˆ2 (q) is Op
2 al (q) .
9. Supercritical Asymptotics; Lack of Consistency A critical issue with both the wavelet based estimator and the moment based ones used in [17], is that the asymptotic properties of the estimators are only valid in a certain range of q-values. For the wavelet estimators, we require q < q ∗ for consistency and for the asymptotic normality results we require 2q < q ∗ . We now show that the range q > q ∗ is uninformative for our estimators and in fact our estimators are misleading when extended to inference for values beyond q ∗ . A reliable estimate of q ∗ would be valuable information. In place of such an estimate it is likely that a graphical procedure is possible based on the following.
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
25
0.0 0.2 0.4 0.6 0.8 1.0 1.2
q = 8 Z/max.Z shape = 10
10 12 14 16 18
2
4
6
8
10
Figure 2. Density plots of Z(q, l)/Z ∨ (q, l). Let τˆi∨ (q) (i = 1, 2) have the same definition as τˆi (q) except that sum is replaced by max. Thus we can define by analogy with (4.3) ∨
Z (q, l) =
l _Y j|l i=1
W (j|i)q |2W (j|l, 0) − 1|q .
Note that q Z ∨ (q, l) = Z ∨ (1, l) .
For large values of q, namely for q ≥ q ∗ , Z(q, l) is sufficiently well approximated by its largest summand Z ∨ (q, l). Figure 2 presents a density plot of simulated values of Z(q, l)/Z ∨ (q, l) as the depth l increases from 10 to 18; note the densities concentrate most mass around the point 1. The cascade generator is a beta distribution with shape parameter 10. Based on the idea of approximating Z(q, l) by Z ∨ (q, l), since log Z ∨ (q, l) = q log Z ∨ (1, l) is linear in q, we anticipate that τˆ1 (q) should also be linear in q rendering τˆ1 (q) largely uninformative for inference purposes in the q ≥ q ∗ –region. A rough estimate of q ∗ would be provided by the q-value where the plots of τˆ1 (q) starts to look linear. Computer simulations offer strong support for these remarks. Figure 3 shows overlaid simulated values for τˆi (q), τˆi∨ (q), i = 1, 2 for large values of q. In the range of q–values beyond q ∗ ≈ 3.3, it is remarkable how linear the plots for τˆ1 (q) and τˆ1∨ (q) look and also how closely τˆ1∨ (q) approximates τˆ1 (q). Note the values in the plots have been multiplied by -1 to make the plots increasing and the cascade generator is a beta distribution with shape parameter 1. We now assume that q ∗ < ∞ and examine this supercritical phenomenon when q ≥ q ∗ in more detail. We will prove the asymptotic linearity of the estimator τˆ1 (q) for q ≥ q ∗ . In particular, the estimator τˆ1 (q) is not consistent when q > q ∗ , and neither is the estimator τˆ2 (q).
26
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
4
levels = 16, q.step = 0.5, shape = 1
3
tau.1 tau.2 theory
-1
0
1
tau(q) 2
max.1 max.2
0
2
4
6
8
10
q Figure 3. Plots of τˆi (q), τˆi∨ (q) for q∗ ≈ 2.4. We start by introducing new notation. Let (9.1)
l
U (q, l) = c(q) M (q, l) =
l XY
W (j|i)q ,
j|l i=1
(9.2) and define, for q > 0, (9.3)
U ∗ (l) = max j|l
l Y
W (j|i),
i=1
l→∞
m(q) = lim inf l→∞
l ≥ 1,
1 log2 U (q, l) , l
m(q) = lim sup
1 log2 U (q, l) l
as well as (9.4)
m∗ = lim sup l→∞
m∗ = lim inf l→∞
q > 0, l ≥ 1 ,
1 log2 U ∗ (l) , l 1 log2 U ∗ (l) . l
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
27
It is immediate that for all q > 0 and 0 ≤ θ ≤ q (9.5)
(U ∗ (l))q ≤ U (q, l) ≤ (U ∗ (l))θ U (q − θ, l) ≤ 2l (U ∗ (l))q .
In particular, for every q > 0
m(q) − 1 ≤ qm∗ ≤ m(q) ,
(9.6)
m(q) − 1 ≤ qm∗ ≤ m(q)
almost surely. Note that it follows from Proposition 6.1 that for 0 < q < q ∗ (9.7)
m(q) = m(q) = τ (q) .
Since by the triangle inequality for all 0 < ρ < 1 and q > 0 (U (q, l))ρ ≤ U (ρq, l) ,
(9.8) we see that (9.9)
m(ρq) ≥ ρm(q) .
For a q ≥ q ∗ and 0 < ρ < q ∗ /q we hence get 1 1 m(q) ≤ m(ρq) = τ (ρq) , ρ ρ and letting ρ ↑ q ∗ /q we conclude that for every q ≥ q ∗ (9.10)
m(q) ≤ q
τ (q ∗ ) . q∗
On the other hand, it follows from (9.5) that for all q > 0 and 0 ≤ θ ≤ q m(q) ≤ θm∗ + m(q − θ) .
Using (9.6) we obtain from here
m(q2 ) + m(q1 − θ) q2 for all q1 , q2 > 0 and 0 ≤ θ ≤ q1 . In particular, if 0 < q1 < q ∗ , then for every 0 < q3 < q1 we choose θ = q1 − q3 and conclude, using (9.7), that m(q1 ) ≤ θ
Therefore, for all q > 0
m(q1 ) − m(q3 ) τ (q1 ) − τ (q3 ) m(q2 ) ≥ = . q2 q1 − q3 q1 − q3 m(q) ≥ sup τ ′ (p) . q 0 0 .
Let 0 < ǫ < 1. Note that it follows from (9.14) that for all l large enough, 1 ∗ ∗ (9.17) P U ∗ (l) ≥ 2(1−ǫ)lτ (q )/q ≥ . 2 For l ≥ 1 let n o Nl = card j|l : W (j|i) ≥ θ for all i = 1, . . . , l .
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
29
By definition N0 = 1. Observe that for all l ≥ 0 Nl+1 = Nl + Ml , where, given N0 , N1 , . . . , Nl , the distribution of Ml is Binomial with parameters Nl and p2 . Therefore, (Nl ) is a supercritical branching process with progeny mean m = 1 + p2 > 1 and extinction probability 0. By Theorem I.10.3 of [1], page 30, lim
(9.18)
l→∞
Nl ˆ > 0 a.s. . =N (1 + p2 )l
Let now 0 < δ < 1. It follows by the definition of (Nl ) that for every l ≥ 1 q (l) Z(l, q) ≥ θ [δl]q max Uk∗ l − [δl] |2Wk − 1|q , (9.19) k=1,... ,N[δl]
where
and
Uk∗ l − [δl] , k ≥ 1 are iid with the law of U ∗ l − [δl] (l)
(Wk , k ≥ 1) are iid with the law of W .
The two sequences are independent, and also independent of N[δl] . All the random variables defined above can be assumed to be defined, for all l and k, on the same probability space (Ω, F, P ). We introduce several events. Let d = (1 + p2 )1/2 > 1. Put n o Ω1 = Nl ≥ dl for all l large enough . It follows from (9.18) that P (Ω1 ) = 1. Let, further, (l) Ω2
=
[δl] d[ n
k=1
o ∗ ∗ (l) |2Wk − 1| ≥ θ and Uk∗ l − [δl] ≥ 2(1−ǫ)(l−[δl])τ (q )/q , (l)
l ≥ 1. Note that by (9.17) we have P (Ω2 ) ≥ 1 − e−cδl for some c > 0 and all l ≥ 1, and so letting (l)
Ω2 = lim inf Ω2 , l→∞
we see by Borel-Cantelli lemma that P (Ω2 ) = 1. Therefore, P (Ω1 ∩ Ω2 ) = 1 as well. However, for every ω ∈ Ω1 ∩ Ω2 we have by (9.19) Z(l, q) ≥ θ [δl]q 2q(1−ǫ)(l−[δl])τ (q
∗ )/q ∗
θq
for all l large enough, which implies that mZ (q) ≥ qδ log2 θ + (1 − ǫ)(1 − δ)q Letting δ → 0 and ǫ → 0 we conclude that (9.20)
mZ (q) ≥ q
τ (q ∗ ) . q∗
Now the statement (9.15) follows from (9.16) and (9.20).
τ (q ∗ ) a.s. . q∗
30
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
Finally, it follows from part (iv) of Proposition 5.2 that τ (q) > q
τ (q ∗ ) q∗
for all q > q ∗ . Hence, the estimator τˆ1 (q, l) is not a consistent estimator of τ (q) if q > q ∗ . Here is an immediate corollary. Corollary 9.1. The estimator τˆ2 (q, l) is not a (strongly) consistent estimator of τ (q) if q > q ∗ . Proof. Notice that for every l ≥ 1,
l−1
τˆ1 (q, l) =
1X τˆ2 (q, j) , l j=0
where Z(q, 0) = 1. Therefore, if for some q > q ∗ τˆ2 (q, l) → τ (q) a.s. as l → ∞, then so does τˆ1 (q, l), which contradicts Theorem 9.1. An estimator related to τˆ2 (q, l) is τˆ3 (q, l) = log2 Since
U (q, l + 1) U (q, l)
:= log2 R(q, l), l ≥ 1 . l−1
(9.21)
1X 1 τˆ3 (q, j) , log2 U (q, l) = l l j=0
where U (q, 0) = 1, (9.13) and the same argument as that of Corollary 9.1 shows that τˆ3 (q, l) is not a strongly consistent estimator of τ (q) if q > q ∗ (even though it is a strongly consistent estimator of τ (q) if q < q ∗ ). We can say more, however. Note that 0 ≤ R(q, l) ≤ 2 for all q and l. Furthermore, ER(q, l) = c(q) = 2τ (q) for all q and l . Therefore, if for some q > q ∗ , τˆ3 (q, l) converges a.s. to some limit τ3 (q) as l → ∞, then the 2τ3 (q) must have a finite expectation equal to 2τ (q) . On the other hand, by (9.13) and (9.21) we must have τ3 (q) equal to qτ (q ∗ )/q ∗ a.s.. This contradiction shows that τˆ3 (q, l) cannot converge a.s. as l → ∞ if q > q ∗ . We conjecture that the same is true for τˆ2 (q, l), in the sense that it does not converge a.s. as l → ∞ if q > q ∗ . A possibility is that τˆ2 (q, l) converges in probability, and is weakly consistent for q > q ∗ . Whether or not this is true remains an open question. 10. Concluding Remarks While Ossiander and Waymire’s estimator for τ (q) is consistent for random cascades, we also check empirically by simulation that it is an appropriate time domain method for conservative cascades. By time domain estimator we mean ! X 1 |µ∞ (I(j|l))|q . τˆtime (q) = log2 l j|l
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
31
levels = 16, q.step = 0.1, instances = 50, shape = 1
-1.0
-0.5
tau(q) 0.0 0.5
1.0
tau.1 tau.2 theory time.one
0.0
0.5
1.0
1.5 q
2.0
2.5
3.0
Figure 4. Plots of the two wavelet estimators and the time domain estimator for τ (q) with q < q∗ ≈ 3.3. We show in Figure 4 that the time domain estimator gives equally good results compared with the two wavelet estimators. The cascade generator is a beta distribution with shape parameter 1. The plot is for q values below q ∗ ≈ 3.3. Note that the τ values are multiplied by -1. One of the advantages of the wavelet method is its ability to filter deterministic trends because different wavelet families have different vanishing moments; i.e., they are orthogonal to low degree polynomials. The Haar wavelets are “blind” to additive constants. Figure 5 illustrates the failure of the time domain method to cope with the presence of an additive constant. The cascade is generated with a beta distribution of shape parameter 1 then a fixed constant 0.1 is added to the cascade. The wavelet estimators give the same values regardless of the presence of the additive constant. Do our wavelet methods work with other wavelet families. One reason for using other wavelets is that the Haar wavelets have only one vanishing moment and can remove only an additive constant. Other wavelet families with higher vanishing moments can remove higher degree deterministic trends. Empirical simulated evidence (shown in Figure 6) suggests that other wavelets do indeed work. Figure 6 shows that τˆ1 works quite well in the case of the D4 wavelet and the presence of an additive linear trend. Note that the D4 wavelet has four vanishing moments (i.e., it is blind to cubic polynomials). The time domain method performs poorly as does τˆ2 . Theoretical investigations are necessary to confirm the validity of the wavelet method for wavelets other than the Haar.
32
S. RESNICK, G. SAMORODNITSKY, A. GILBERT, AND W. WILLINGER
tau(q) 0.0 0.5
1.0
levels = 16, q.step = 0.1, instances = 50, shape = 1, trend added
-1.0
-0.5
tau.1 tau.2 theory time.trend.one
0.0
0.5
1.0
1.5 q
2.0
2.5
3.0
Figure 5. Plots of the wavelet and time domain estimators for a cascade with an additive constant.
References [1] K. Athreya and P. Ney. Branching Processes. Springer-Verlag, New York, 1972. [2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. [3] A. Feldmann, A. C. Gilbert, and W. Willinger. Data networks as cascades: Investigating the multifractal nature of Internet WAN traffic. In Proc. of the ACM/SIGCOMM’98, pages 25–38, Vancouver, B.C., 1998. [4] U. Frisch and G. Parisi. Fully developed turbulence and intermittancy. In M. Ghil, editor, Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland, Amsterdam, 1985. [5] A.C. Gilbert, W. Willinger, and A. Feldmann. Scaling analysis of conservative cascades, with applications to network traffic. IEEE Transactions on Information Theory, 45(3):971–991, 1999. [6] V. K. Gupta and E. C. Waymire. A statistical analysis of mesoscale rainfall as a random cascade. Journal of Applied Meteorology, 32:251–267, 1993. [7] Richard Holley and Edward C. Waymire. Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Probab., 2(4):819–845, 1992. [8] J.P. Kahane and J. Peyri´ere. Sur certaines martingales de b. mandelbrot. Advances in Mathematics, 22:131– 145, 1976. [9] J.F.C. Kingman. The first birth problem for an age–dependent branching process. The Annals of Probability, 3:790–801, 1975. [10] A.N. Kolmogorov. Local structure of turbulence in an incompressible liquid for very large Reynolds numbers. C.R. Doklady Acad. Sci. URSS (N.S.), 30:299–303, 1941. [11] H. Mahmood. Evolution of Random Search Trees. Wiley, New York, 1992.
WAVELET ANALYSIS OF CONSERVATIVE CASCADES
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levels = 16, q.step = 0.1, instances = 50, shape = 1, trend added, D4
-1
0
tau(q)
1
2
tau.1 tau.2 theory time.trend.one
0.0
0.5
1.0
1.5 q
2.0
2.5
3.0
Figure 6. Plots of the wavelet and time domain estimators using D4 wavelets and in the presence of an additive linear trend.
[12] B. B. Mandelbrot. Intermittant turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. Journal of Fluid Mechanics, 62:331–358, 1974. [13] B. B. Mandelbrot. Limit Lognormal Multifractal Measures. In Gotsman, Ne’eman, and Voronel, editors, The Landau Memorial Conference, pages 309–340, Tel Aviv, 1990. [14] B.B. Mandelbrot. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer-Verlag, New York, 1998. [15] C. Meneveau and K. R. Sreenivasan. Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett., 59:1424–1427, 1987. [16] J. Neveu. Discrete-Parameter Martingales, volume 10 of North-Holland Mathematical Library. North Holland, Amsterdam, 1975. Translated from the French original by T.P. Speed. [17] M. Ossiander and E. Waymire. Statistical estimation for multiplicative cascades. To appear; available from {ossiand,waymire}@math.orst.edu, 1999. [18] S.I. Resnick. A Probability Path. Birkh¨ auser, Boston, 1998. [19] R. H. Riedi and J. Levy-Vehel. Tcp traffic is multifractal: A numerical study. Preprint, 1997. [20] Brent M. Troutman and Aldo V. Vecchia. Estimation of R´enyi exponents in random cascades. Bernoulli, 5(2):191–207, 1999.
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School of Operations Research and Industrial Engineering and Department of Statistical Science, Cornell University, Ithaca, NY 14853 E-mail address: [email protected], [email protected]
AT&T Labs–Research, 180 Park Avenue, Florham Park, NJ 07932 E-mail address: [email protected], [email protected]
