Invariance principles for fractionally integrated nonlinear processes
The University of Chicago Department of Statistics TECHNICAL REPORT SERIES
Invariance Principles for Fractionally Integrated Nonlinear Processes Xiaofeng Shao, Wei Biao Wu
TECHNICAL REPORT NO. 561
December 6, 2005
The University of Chicago Department of Statistics 5734 S. University Avenue Chicago, IL 60637
___________________________________ *The work is supported in part by NSF grant DMS-0478704.
IMS Lecture Notes–Monograph Series
Invariance principles for fractionally integrated nonlinear processes∗ Xiaofeng Shao, Wei Biao Wu University of Chicago Abstract: We obtain invariance principles for a wide class of fractionally integrated nonlinear processes. The limiting distributions are shown to be fractional Brownian motions. Under very mild conditions, we extend earlier ones on long memory linear processes to a more general setting. The invariance principles are applied to the popular R/S and KPSS tests.
1. Introduction Invariance principles (or functional central limit theorems) play an important role in econometrics and statistics. For example, to obtain asymptotic distributions of unitroot test statistics, researchers have applied invariance principles of various forms; see Phillips (1987), Sowell (1990) and Wu (2006) among others. The primary goal of the paper is to establish invariance principles for a class of fractionally integrated nonlinear processes. Let the process ut = F (· · · , εt−1 , εt ), t ∈ Z,
(1.1)
where εt are independent and identically distributed (iid) random variables and F is a measurable function such that ut is well-defined. Then ut is stationary and causal. Let d ∈ (−1/2, 1/2) and define Type I fractional I(d) process Xt by (1 − B)d (Xt − µ) = ut , t ∈ Z,
(1.2)
where µ is the mean and B is the backward shift operator: BXt = Xt−1 . The Type II I(d) fractional process is defined as (1 − B)d (Yt − Y0 ) = ut 1(t ≥ 1).
(1.3)
where Y0 is a random variable whose distribution is independent of t. There is a recent surge of interest in Type II processes [Robinson and Marinucci (2001), Phillips and Shimotsu (2004)] and it arises naturally when the processes start at a given time point. The framework (1.1) includes a very wide class of processes [Wiener (1958), Rosenblatt (1971), PriestleyP (1988), Tong (1990), Wu (2005a), Tsay ∞ (2005)]. It includes linear processes ut = j=0 bj εt−j as a special case. It also includes a large class of nonlinear time series models, such as bilinear models, threshold models and GARCH type models [Wu and Min (2005), Shao and Wu (2005)]. Recently, fractionally integrated autoregressive and moving average models (FARIMA) with GARCH innovations have attracted much attention in financial time series modeling [see Baillie et al. (1996), Hauser and Kunst (1998), Lien and ∗ The
work is supported in part by NSF grant DMS-0478704. AMS 2000 subject classifications: Primary 60F17; secondary 62M10 Keywords and phrases: Fractional integration, Long memory, Nonlinear time series, Weak convergence
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Tse (1999) among others]. The FARIMA-GARCH model naturally fits into our framework and partially motivates us to study asymptotic properties of its partial sums. Most of the results in the literature assume {ut } to be either iid or linear processes. Recently, Wu and Min (2005) established an invariance principle under (1.2) when d ∈ [0, 1/2). The literature seems more concentrated on the case d ∈ (0, 1/2). Part of the reason is that this case corresponds to long memory and it appears in various areas such as finance, hydrology and telecommunication. When Pt Pt d ∈ (1/2, 1), the process is non-stationary and it can be defined as s=1 Xs or s=1 Ys , where Xs and Ys are Type I and Type II I(d − 1) processes, respectively. Empirical evidence of d ∈ (1/2, 1) has been found by Byers et al. (1997) in the poll data modeling and Kim (2000) in macroeconomics time series. Therefore the study of partial sums of I(d), d ∈ (−1/2, 0) is also of interest since it naturally leads to I(d) processes, d ∈ (1/2, 1). In fact, our results can be easily extended to the process with fractional index p + d, p ∈ N, d ∈ (−1/2, 0) ∪ (0, 1/2) [cf. Corollary 2.1]. The study of invariance principle has a long history. Here we only mention some representatives: Davydov (1970), Mcleich (1977), Gorodetskii (1977), Hall and Heyde (1980), Phillips and Solo (1992), Davidson and De Jong (2000), De Jong and Davidson (2000) and the references cited therein. Most of them deal with Type I processes. Recent developments for Type II processes can be found in Marinucci and Robinson (1999a), Wang et al. (2002) and Hosoya (2005) among others. The paper is structured as follows. Section 2 presents invariance principles for both types of processes. Section 3 considers limit distributions of tests of long memory under mild moment conditions. Technical details are given in the appendix. 2. Main Results We first define two types of fractional Brownian motions. For Type I fractional Brownian motion, let d ∈ (−1/2, 1/2) and Z ∞ 1 IBd (t) = [{(t − s)+ }d − {(−s)+ }d ]dIB(s), t ∈ R, A(d) −∞ where (t)+ = max(t, 0), IB(s) is a standard Brownian motion and A(d) =
1 + 2d + 1
Z
∞
[(1 + s)d − sd ]2 ds
1/2 .
0
Type II fractional Brownian motion {Wd (t), t ≥ 0}, d > −1/2, is defined as Wd (0) = 0, Wd (t) = (2d + 1)1/2
Z
t
(t − s)d dIB(s).
0
The main difference of IBd (t) and Wd (t) lies in the prehistoric treatment. See Marinucci and Robinson (1999b) for a detailed discussion of the difference between them. Pm Here we are interested in the weak convergence of the partial sums T = m i=1 Xi Pm and Tm = i=1 Yi . Let D[0, 1] be the space of functions on [0, 1] which are right continuous and have left-hand limits, endowed with the Skorohod topology (Billingsley, 1968). Denote weak convergence by ”⇒”. For a random variable X, write X ∈ Lp (p > 0) if kXkp := [E(|X|p )]1/p < ∞ and k · k = k · k2 . Let Ft = (· · · , εt−1 , εt ) be the shift process. Define the projections imsart-lnms ver.
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Pk by Pk X = E(X|Pk ) − E(X|Pk−1 ), X ∈ L1 . For two sequences (an ), (bn ), denote by an ∼ bn if an /bn → 1 as n → ∞. The symbols “→D ” and “→P ” stand for convergence in distribution and in probability, respectively. The symbols OP (1) and oP (1) signify being bounded in probability and convergence to zero in probability. Let N (µ, σ 2 ) be a normal distribution with mean µ and variance σ 2 . Hereafter we assume without loss of generality that E(ut ) = 0, µ = 0 and Y0 = 0. Let {ε0k , k ∈ Z} be an iid copy of {εk , k ∈ Z} and Fk∗ = (F−1 , ε00 , ε1 , . . . , εk ). Theorems 2.1 and 2.2 concern Type I and II processes respectively. Using the continuous mapping theorem, Theorems 2.1 and 2.2 imply Corollary 2.1 which deals with general fractional processes with higher orders. For d ∈ (−1/2, 0), an undesirable feature of our results is that the moment condition depends on d. However, this seems to be necessary; see Remark 4.1. Similar conditions were imposed in Sowell (1990) and Wang et al. (2003). Theorem 2.2 extends early results by Akonom and Gourieroux (1987), Tanaka (1999) and Wang et al. (2002), who assumed ut to be either iid or linear processes. See Marinucci and Robinson (1999a) and Hosoya (2005) for a multivariate extension. Theorem 2.1. Let ut ∈ Lq , where q > 2/(2d + 1) if d ∈ (−1/2, 0) and q = 2 if d ∈ (0, 1/2). Assume ∞ X
kP0 uk kq < ∞.
(2.1)
k=0
Then
P∞
k=j
Pj uk → ζj in Lq and, if kζ0 k > 0,
Tbntc A(d)kζ0 k . ⇒ κ1 (d)IBd (t) in D[0, 1], where κ1 (d) = Γ(d + 1) nd+1/2
(2.2)
Remark 2.1. Note that kζ0 k2 = 2πfu (0), where fu (·) is the spectral density function of {ut }; see Wu (2005b) and Wu and Min (2005) for the details. Theorem 2.2. Under the conditions of Theorem 2.1, for type II processes, we have Tbntc
nd+1/2
⇒ κ2 (d)Wd (t) in D[0, 1], where κ2 (d) =
kζ0 k(2d + 1)−1/2 . Γ(d + 1)
(2.3)
Corollary 2.1. Let ut satisfy conditions in Theorem 2.1; let d ∈ (0, 1/2) ∪ ˜ t by (1 − B)p+d X ˜ t = ut . Then (−1/2, 0) and p ∈ N. [a] Define the process X −(d+p−1/2) ˜ Xbntc ⇒ κ1 (d)IBd,p (t) in D[0, 1]; (i). n Pbntc ˜ −(d+p+1/2) (ii). n j=1 Xj ⇒ κ1 (d)IBd,p+1 (t) in D[0, 1]; Rt Pbntc ˜ 2 2 −2(d+p) 2 (iii). n j=1 Xj ⇒ κ1 (d) 0 [IBd,p (s)] ds in D[0, 1]. Here IBd,p (t) is defined as IBd,p (t) =
R t R tp−1 0
0
···
R t2 0
IBd (t), IBd (t1 )dt1 dt2 · · · dtp−1 ,
p = 1, p ≥ 2.
[b] Define the process Y˜t by (1 − B)p+d Y˜t = ut 1(t ≥ 1). Then similarly (i). n−(d+p−1/2) Y˜bntc ⇒ κ2 (d)Wd+p−1 (t) in D[0, 1]; Pbntc (ii). n−(d+p+1/2) j=1 Y˜j ⇒ κ2 (d)Wd+p (t) in D[0, 1]; Rt Pbntc ˜ 2 (iii). n−2(d+p) Y ⇒ κ2 (d) [Wd+p−1 (s)]2 ds in D[0, 1]. j=1
j
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0
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We now discuss condition (2.1). Let gk (F0 ) = E(uk |F0 ) and δq (k) = kgk (F0 ) − gk (F0∗ )kq . Then δq (k) measures the contribution of ε0 in predicting uk . In Wu (2005a) it is called the predictive dependence measure. Since kP0 uk kq ≤ δq (k) ≤ 2kP0 uk kq (Wu, 2005a), (2.1) is equivalent to the q-stability condition (Wu, 2005a) P ∞ k=0 δq (k) < ∞, which suggests short-range dependence in that the cumulative contribution of ε0 in predicting future values of uk is finite. For a variety of nonlinear time series, δq (k) = O(ρk ) for some ρ ∈ (0, 1). The latter is equivalent to the geometric moment contraction (GMC) [Wu and Shao (2004), Wu and Min (2005)]. Shao and Wu (2005) verified GMC for GARCH(r, s) model and its asymmetric variants and showed P that the GMC property is preserved under ARMA filter. In P∞ ∞ the special case ut = k=0 bk εt−k , (2.1) holds if k=0 |bk | < ∞ and ε1 ∈ Lq . 3. Applications There have been a large amount of work on test of long memory under short memory null hypothesis, i.e. I(0) versus I(d), d ∈ (0, 1/2). For example, Lo (1991) introduced modified R/S test statistics, which admits the following form: k k X X 1 ¯n) , ¯ n ) − min (Xj − X (Xj − X max Qn = 1≤k≤n wn,l 1≤k≤n j=1
j=1
Pn −1
¯n = n where X j=1 Xj is the sample mean and wn,l is the long run variance estimator of Xt . Following Lo (1991), 2 wn,l
l n X 1X j 2 ¯ = 1− (Xj − Xn ) + 2 γˆj , n j=1 l+1 j=1
(3.1)
Pn−j ¯ n )(Xi+j − X ¯ n ), 0 ≤ j < n. The form (3.1) is equivalent where γˆj = n1 i=1 (Xi − X to the nonparametric spectral density estimator of {Xt } evaluated at zero frequency with Bartlett window (up to a constant factor). Here the bandwidth satisfies l = l(n) → ∞ and l/n → 0, as n → ∞.
(3.2)
Lee and Schmidt (1996) applied the KPSS test (Kwiatkowski et al. 1992) for I(0) versus I(d), d ∈ (−1/2, 0) ∪ (0, 1/2). The test statistics has the form: 2 k n X X 1 ¯ n ) (Xj − X Kn = 2 2 wn,l n j=1 k=1
2 with wn,l given by (3.1). Lee and Schmidt showed that the test is consistent against fractional alternatives and derived its asymptotic distribution under the assumption that ut are iid normal random variables. Giraitis et al. (2003) investigated the theoretical performance of various forms of nonparametric tests under both short memory hypotheses and long memory alternatives. In a quite general setting, we obtain asymptotic distributions of R/S and KPSS test statistics under fractional alternatives.
Theorem 3.1. Suppose that Xt is generated from (1.2) and ut satisfies (2.1) with some q > max(2, 2/(2d + 1)). Assume (3.2). Then 2 l−2d wn,l →P κ21 (d).
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Consequently, we have ld nd+1/2
˜ d (t) − inf IB ˜ d (t), Qn →D sup IB 0≤t≤1
0≤t≤1
(3.4)
˜ d (t) is the fractional Brownian bridge, i.e. IB ˜ d (t) = IBd (t) − tIBd (1), and where IB l2d Kn →D n2d
Z
1
˜d (t))2 dt. (B
(3.5)
0
Remark 3.1. For d ∈ (0, 1/2), Giraitis et al. (2003) obtained (3.4) under the joint cumulant summability condition sup
n X
|cum(X0 , Xh , Xr , Xs )| = O(n2d ).
(3.6)
h r,s=−n
For linear processes, (3.6) can be verified. But for nonlinear fractional processes (1.2), it seems hard to directly verify (3.6). In contrast, we only need to impose q-th (q > 2) moment condition when d ∈ (0, 1/2). Our dependence condition (2.1) can be easily verified for various nonlinear time series models [cf. Wu and Min (2005) and Shao and Wu (2005)]. 4. Appendix Lemma 4.1. Let ai = i−β `(i), i ≥ 1, where ` is a slowly varying function and β > 1/2; let q > (3/2 − β)−1 if 1 < β < 3/2 and q = 2 if 1/2 < β < 1; let σn = A(1 − β)n3/2−β `(n)/(1 P − β) and σ ˜n = (3 − 2β)−1/2 n3/2−β `(n)/(1 − β). ∞ ◦ Assume either (1 ) 1 < β < 3/2, i=0 ai = 0 or (2◦ ) 1/2 < β < 1. Further assume that the martingale differences ζj = F (. . . , εj−1 , εj ), j ∈ Z,
(4.1)
P∞ Pj−1 Pi satisfy ζj ∈ Lq . Let Yj = i=0 ai ζj−i , Yj = i=0 ai ζj−i , Si = j=1 Yj , Si = Pi −1 ⇒ ˜n−1 Sbntc j=1 Yj . Then we have [a] σn Sbntc ⇒ kζ0 kIB3/2−β (t) in D[0, 1] and [b] σ kζ0 kW3/2−β (t) in D[0, 1]. Proof of Lemma 4.1. [a] Consider (1◦ ) first. For the finite dimensional convergence, we shall apply the Cramer-Wold device. Fix 0 ≤ t1 < t2 ≤ 1 and let m1 = bnt1 c Pi and m2 = bnt2 c. Let Ai = j=0 aj if i ≥ 0 and Ai = 0 if i < 0. For λ, µ ∈ R let λ(Am1 −m2 +l − Al−m2 ) + µ(Al − Al−m2 ) , σn 2 σλµ = kζ0 k2 [λ2 t3−2β + µ2 t3−2β + λµ(t3−2β + t3−2β − (t2 − t1 )3−2β )]. 1 2 1 2 P∞ Then (λSm1 + µSm2 )/σn = l=0 cn,l ζm2 −l has martingale difference summands and we can apply the martingale central limit theorem. By Karamata’s Theorem, P∞ An = − j=n+1 aj ∼ n1−β `(n)/(β − 1). Elementary calculations show that cn,l
=
∞ X
2 c2n,l → σλµ and sup |cn,l | → 0 as n → ∞.
l=0
imsart-lnms ver.
(4.2)
l≥0
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2 Let Vl = E(ζm |Fm2 −l−1P ). By the argument in the proof of Theorem 1 in Hannan 2 −l ∞ 2 (1973), (4.2) implies that l=0 c2n,l Vl → σλµ in L1 . For completeness we prove it 0 00 here. Let ω > 0 be fixed, Vl = Vl 1Vl ≤ω and Vl = Vl − Vl0 . By (4.2),
2 ∞ ∞
X
X
0 c2n,l c2n,l0 |cov(V00 , Vl−l c2n,l (Vl0 − EVl0 ) ≤ lim sup lim sup 0 )| = 0
n→∞ n→∞ 0 l,l =0
l=0
since
limk→∞ cov(V00 , Vk0 )
= 0. Therefore, using Vl = Vl0 + Vl00 , again by (4.2), ∞ ∞ X X 2 2 00 00 cn,l (Vl − EVl ) lim sup E cn,l (Vl − EVl ) ≤ lim sup E n→∞ n→∞ l=0 l=0 ∞ X ≤ 2 lim sup c2n,l EVl00 → 0 as ω → ∞. n→∞
l=0
P∞
2 2 l=0 E{|cn,l ζm2 −l |1|cn,l ζm2 −l |≥δ }
Under (4.2), for any δ > 0, → 0. So the finite dimensional convergence holds. By Proposition 4 of Dedecker and Doukhan (2003), kSn k2q ≤ 2qkζ0 k2q
∞ X
(Aj − Aj−n )2 = O(σn2 ).
j=1
By Theorem 2.1 of Taqqu (1975), the tightness follows. (2◦ ) Note that kSn k ∼ kζ0 kσn , the conclusion similarly follows. [b] The finite dimensional convergence follows in the same manner as [a]. For the tightness, let 1 ≤ m1 < m2 ≤ n, by Proposition 4 in Dedecker and Doukhan (2003), kSm − Sm k2 ≤ 2qkζ0 k2q 2 1 q
m 2 −1 X
2 (Aj − Aj−(m2 −m1 ) )2 = O(˜ σm ). 2 −m1
j=0
With the above inequality, using the same argument as in Theorem 2.1 of Taqqu (1975), we have for any 0 ≤ t1 ≤ t ≤ t2 ≤ 1, there exists a generic constant C (independent of n, t1 , t, t2 ), such that for β ∈ (1/2, 1), E|Sbntc − Sbnt ||Sbnt − Sbntc | ≤ Cσ ˜n2 (t2 − t1 )3−2β 1c 2c
and for β ∈ (1, 3/2), E|Sbntc − Sbnt |q/2 |Sbnt − Sbntc |q/2 ≤ C σ ˜nq (t2 − t1 )q(3/2−β) . 1c 2c
Thus the tightness follows from Theorem 15.6 in Billingsley (1968).
♦
Remark 4.1. Under (1◦ ) of Lemma 4.1, the moment condition ζj ∈ Lq , q > (3/2 − β)−1 , is optimal. and it can not be reduced to ζj ∈ Lq0 , q0 = (3/2 − β)−1 . Consider the case in which ζi are iid symmetric random variables and P(|ζ0 |q0 ≥ g) ∼ g −1 (log g)−2 as g → ∞. Then ζj ∈ Lq0 . Let `(n) = 1/ log n, n > 3. Elementary calculations show that σn−1 max1≤j≤n |ζj | → ∞ in probability. Let Pi 0 0 0 Yj0 = Yj + ζj − ζj−1 and Si0 = j=1 Yj . Then the coefficients aj of Yj also 0 , satisfy the conditions in Lemma 4.1. The two processes σn−1 Sbntc and σn−1 Sbntc 0 ≤ t ≤ 1, cannot both converge weakly to fractional Brownian motions. If so, since maxj≤n |ζj − ζ0 | ≤ maxj≤n |Sj | + maxj≤n |Sj0 |, we have maxj≤n |ζj | = OP (σn ), contradicting σn−1 max1≤j≤n |ζj | → ∞ in probability. Similar examples are given in Wu and Min (2005) and Wu and Woodroofe (2004). ♦ imsart-lnms ver.
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Pk Proof of Theorem 2.1. Let aj = Γ(j + d)/{Γ(d)Γ(j + 1)}, j P ≥ 0, and Ak = i=0 ai ∞ if k ≥ 0 and 0 if k < 0. Note that β = 1 − d. Then Xt = j=0 aj ut−j . By (2.1), P∞ P P P∞ n n ζj = i=j Pj ui ∈ Lq . Let Mn = i=1 ζi , Sn = j=1 Yj , Yj = i=0 ai ζj−i , Un = P √ n i=1 ui and Rn = Tn − Sn . By Theorem 1 in Wu (2005b), kUn − Mn kq = o( n). By Karamata’s theorem and summation by parts, we have
3m
X
kRm kq ≤ (Ai − Ai−m )(um−i − dm−i )
i=0 q
∞
X
+ (Ai − Ai−m )(um−i − dm−i )
i=3m+1
=
3m X i=1
+
q
√ |(Ai − Ai−m ) − (Ai−1 − Ai−1−m )|o( i) ∞ X
√ |(Ai − Ai−m ) − (Ai−1 − Ai−1−m )|o( i) = o(σm ).
i=3m+1
By Proposition 1 in Wu (2005b),
max |Rm |
m≤2k
q
≤
k X
2(k−r)/q kR2r kq =
r=0
k X
2(k−r)/q o(σ2r ) = o(σ2k ),
r=0
♦
since q > 2/(2d + 1). So the theorem follows from Lemma 4.1. Proof of Theorem 2.2. As in the proof of Theorem 2.1, let Si = . By Karamata’s theorem, − Sm = Tm Rm kRm+l − Rl kq ≤
m X
Pi
j=1
Yj and
p |Aj − Aj−1 |o( j) = o(˜ σm ).
j=1
Again by the maximal inequality [Proposition 1 in Wu (2005b)], k−r 1/q
2X k X
max |Rm kR2r j − R2r (j−1) kqq | ≤
m≤2k
q
r=0
=
k X
j=1
2(k−r)/q o(˜ σ2r ) = o(˜ σ2k ),
r=0
which proves the theorem in view of Lemma 4.1.
♦
Proof of Theorem 3.1. If (3.3) holds, by the continuous mapping theorem, Theorem 2.1 entails (3.4) and (3.5). In the sequel we shall prove (3.3). Note that 2 wn,l
=
n−j n l X 1X 2 2X j ¯ n2 Xj + 1− Xi Xi+j − X n j=1 n j=1 l + 1 i=1 n−j l l X ¯n X ¯ n2 X 2X j 2X j − 1− (Xi + Xi+j ) + 1− (n − j) n j=1 l + 1 i=1 n j=1 l+1
¯ n2 + I2n . =: I1n − X imsart-lnms ver.
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¯ n | = OP (nd−1/2 ), l−2d (X ¯ n2 + |I2n |) = OP ((l/n)1−2d ) = oP (1). Thus it Since |X Pn Pj suffices to show that l−2d I1n →P κ21 (d). Let Vj = i=1 Xi , V˜j = i=n−j+1 Xi , 1 ≤ j ≤ l, then a straightforward calculation shows that I1n = J1n + J2n , where
J1n :=
1 (l + 1)n
n X
i X
i=l+1
2
l
Xj , J2n :=
j=i−l
X 1 (V 2 + V˜j2 ). n(l + 1) j=1 j
oP (l2d ). Corollary 2.1 implies that J2n = OP (l2d+2 /(ln)) =P Pn ∞ −2d 2 It remains to show l J1n →P κ1 (d). Let ζj = i=j Pj ui ∈ Lq ; Mn = i=1 ζi , Pn √ Un = i=1 ui and rn =√supj≥n kUj − Mj k/ j. Then rn → 0 and it is nonincreasing. Let L = bmin{ nl, l(r√l )1/(2β−3) }c. Then l = o(L) and L = o(n). Let Wj,l =
L X
(Ai − Ai−l )uj−i ,
Qj,l =
i=0
L X
(Ai − Ai−l )ζj−i
i=0
and b = bn/(2L)c. Since τm := E|E(ζ02 |F−m ) − E(ζ02 )| → 0 as m → ∞, E|E(Q22Lk,l |F2Lk−2L ) − E(Q22Lk,l )|
≤
L X
(Ai − Ai−l )2 τ2L−i = o(σl2 ). (4.3)
i=0
Let Dk = Q22Lk,l − E(Q22Lk,l |F2Lk−2L ), k = 1, . . . , b. Set Cq = 18q 3/2 (q − 1)−1/2 and q 0 = min(q, 4). By Burkholder’s inequality,
b
X
0 0
Dk ≤ Cq/2 b2/q kDk kq/2 ≤ 2Cq/2 b2/q kQ2Lk,l k2q
k=1
q/2
0
= b2/q O(σl2 ) = o(bσl2 ).
(4.4)
By (4.3) and (4.4), it is easily seen that n X 2 E {Qj,l − E(Q2j,l )} = o(nσl2 ). j=1 Let Sj,l =
Pj
i=j−l+1
(4.5)
Xi . Since l/L → 0,
∞
X
kSj,l − Wj,l k = (Ai − Ai−l )uj−i
i=L+1 " ∞ #1/2 ∞ X X ≤ (Ai − Ai−l )2 kP0 ut k = o(σl ).
(4.6)
t=0
i=L+1
By the definition of L, using summations by parts, we have kWj,l − Qj,l k
≤
L X
√ |(Ai − Ai−l ) − (Ai−1 − Ai−l−1 )|ri i
i=1
≤ r√ l
imsart-lnms ver.
L X √ i=1+ l
√
l X √ √ 2|ai |ri i = o(σl ). 2|ai | i +
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i=1
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By (4.6) and (4.7), kSj,l − Qj,l k = o(σl ). Hence by (4.5), X X n n 2 2 2 E|Sj,l E {Sj,l − E(Qj,l )} ≤ − Q2j,l | + o(nσl2 ) = o(nσl2 ) j=1 j=1 which completes the proof since E(Q2j,l ) ∼ σl2 kζ0 k2 .
♦
REFERENCES
Akonom, J. and Gourieroux, C. (1987). A functional central limit theorem for fractional processes. Discussion paper #8801. Paris: CEPREMAP. Baillie, R. T., Chung, C. F. and Tieslau, M. A. (1996). Analyzing inflation by the fractionally integrated ARFIMA-GARCH model. J. Appl. Econom. 11 23-40. Billingsley, P. (1968). Convergence of Probability Measures. Wiley. Byers, D., Davidson, J. and Peel, D. (1997). Modelling political popularity: an analysis of long-range dependence in opinion poll series. J. Roy. Statist. Soc. Ser. A. 160 471-490. Chung, K. L. (1968). A Course in Probability Theory. Harcourt, Brace and World, New York. Davidson, J. and De Jong, R. M. (2000). The functional central limit theorem and weak convergence to stochastic integrals, II: Fractionally integrated processes. Econom. Theory 16 643-666. De Jong, R. M. and Davidson, J. (2000). The functional central limit theorem and weak convergence to stochastic integrals, I: Weakly dependent processes. Econom. Theory 16 621-642. Davydov, Y. A. (1970). The invariance principle for stationary processes. Theory Probab. Appl. 15 487-498. Dedecker, J. and Doukhan, P. (2003). A new covariance inequality and applications. Stochast. Process. Appl. 106 63-80. ´re, G. (2003). Rescaled Giraitis, L., Kokoszka, P., Leipus, R. and Teyssie variance and related tests for long memory in volatility and levels. J. Econometrics 112 265-294. Gorodetskii, V. V. (1977). On convergence to semi-stable Gaussian process. Theory Probab. Appl. 22 498-508. Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Applications. New York: Academic Press. imsart-lnms ver.
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Hannan, E. J. (1973). Central limit theorems for time series regression. Z. Wahrsch. Verw. Geb 26 157-170. Hauser, M. A. and Kunst, R. M. (1998). Forecasting high-frequency financial data with the ARFIMA-ARCH model, J. Forecasting 20 501-518. Hosoya, Y. (2005). Fractional invariance principle. J. Time Ser. Anal. 26 463-486. Kim, C. S. (2000). Econometric analysis of fractional processes. Unpublished Ph.D. thesis. Yale University. Kwiatkowski, D., Phillips, P. C. B., Schmidt, P. and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? J. Econometrics 54 159-178. Lee, D. and Schmidt, P. (1996). On the power of the KPSS test of stationarity against fractionally-integrated alternatives. J. Econometrics 73 285-302. Lien, D. and Tse, Y. K. (1999). Forecasting the Nikkei spot index with fractional cointegration, J. Forecasting 18 259-273. Lo, A. (1991). Long-term memory in stock market prices. Econometrica 59 12791313. Marinucci, D. and Robinson, P. M. (1999a). Weak convergence of multivariate fractional processes. Stochast. Process. Appl. 86 103-120. Marinucci, D. and Robinson, P. M. (1999b). Alternative forms of fractional Brownian motion. J. Stat. Plan. Infer. 80 111-122. Mcleish, D. L. (1977). On the invariance principle for non-stationary mixingales. Ann. Probab. 5 616-621. Phillips, P. C. B. (1987). Time series regression with a unit root. Econometrica 55 277-301. Phillips, P. C. B. and Shimotsu, K. (2004). Local Whittle estimation in nonstationary and unit root cases. Ann. Statist. 32 656-692. Priestley, M. B. (1988). Nonlinear and Nonstationary Time Series Analysis. Academic Press, London. Robinson, P. M. and Marinucci, D. (2001). Narrow-band analysis of nonstationary processes. Ann. Statist. 27 947-986. Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior. Springer, New York. Shao, X. and Wu, W. B. (2005). Asymptotic spectral theory for nonlinear time series. Preprint. Sowell, F. (1990). The fractional unit root distribution. Econometrica 58 495-505. imsart-lnms ver.
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Tanaka, K. (1999). The nonstationary fractional unit root. Econom. Theory 15 549-582. Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z.Wahrsch. Verw. Geb. 31 287-302. Tong, H. (1990). Non-linear Time Series: A Dynamical System Approach. Oxford University Press. Tsay, R. S. (2005). Analysis of Financial Time Series. Wiley, New York. Wang, Q., Lin, Y. X. and Gulati, C. M. (2002). Asymptotics for general fractionally integrated processes without prehistoric influence. Journal of Applied Mathematics and Decision Sciences 6(4) 255-269. Wang, Q., Lin, Y. X. and Gulati, C. M. (2003). Asymptotics for general fractionally integrated processes with applications to unit root tests. Econom. Theory 19 143-164. Wiener, N. (1958). Nonlinear Problems in Random Theory. MIT press, MA. Wu, W. B. (2005a). Nonlinear system theory: another look at dependence. Proc. Natl. Acad. Sci. 102 14150-14154. Wu, W. B. (2005b). A strong convergence theory for stationary processes. Preprint. Wu, W. B. (2006). Unit root testing for functional of linear processes. Econom. Theory 22 1-14. Wu, W. B. and Min, W. (2005). On linear processes with dependent innovations. Stochast. Process. Appl. 115 939-958. Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Prob. 41 425-436. Wu, W. B. and Woodroofe, M. (2004). Martingale approximations for sums of stationary processes. Ann. Probab. 32 1674-1690. DEPARTMENT OF STATISTICS UNIVERSITY OF CHICAGO 5734 S. UNIVERSITY AVE, CHICAGO, IL 60637 E-mail: shao,[email protected]
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Invariance Principles for Fractionally Integrated Nonlinear Processes Xiaofeng Shao, Wei Biao Wu
TECHNICAL REPORT NO. 561
December 6, 2005
The University of Chicago Department of Statistics 5734 S. University Avenue Chicago, IL 60637
___________________________________ *The work is supported in part by NSF grant DMS-0478704.
IMS Lecture Notes–Monograph Series
Invariance principles for fractionally integrated nonlinear processes∗ Xiaofeng Shao, Wei Biao Wu University of Chicago Abstract: We obtain invariance principles for a wide class of fractionally integrated nonlinear processes. The limiting distributions are shown to be fractional Brownian motions. Under very mild conditions, we extend earlier ones on long memory linear processes to a more general setting. The invariance principles are applied to the popular R/S and KPSS tests.
1. Introduction Invariance principles (or functional central limit theorems) play an important role in econometrics and statistics. For example, to obtain asymptotic distributions of unitroot test statistics, researchers have applied invariance principles of various forms; see Phillips (1987), Sowell (1990) and Wu (2006) among others. The primary goal of the paper is to establish invariance principles for a class of fractionally integrated nonlinear processes. Let the process ut = F (· · · , εt−1 , εt ), t ∈ Z,
(1.1)
where εt are independent and identically distributed (iid) random variables and F is a measurable function such that ut is well-defined. Then ut is stationary and causal. Let d ∈ (−1/2, 1/2) and define Type I fractional I(d) process Xt by (1 − B)d (Xt − µ) = ut , t ∈ Z,
(1.2)
where µ is the mean and B is the backward shift operator: BXt = Xt−1 . The Type II I(d) fractional process is defined as (1 − B)d (Yt − Y0 ) = ut 1(t ≥ 1).
(1.3)
where Y0 is a random variable whose distribution is independent of t. There is a recent surge of interest in Type II processes [Robinson and Marinucci (2001), Phillips and Shimotsu (2004)] and it arises naturally when the processes start at a given time point. The framework (1.1) includes a very wide class of processes [Wiener (1958), Rosenblatt (1971), PriestleyP (1988), Tong (1990), Wu (2005a), Tsay ∞ (2005)]. It includes linear processes ut = j=0 bj εt−j as a special case. It also includes a large class of nonlinear time series models, such as bilinear models, threshold models and GARCH type models [Wu and Min (2005), Shao and Wu (2005)]. Recently, fractionally integrated autoregressive and moving average models (FARIMA) with GARCH innovations have attracted much attention in financial time series modeling [see Baillie et al. (1996), Hauser and Kunst (1998), Lien and ∗ The
work is supported in part by NSF grant DMS-0478704. AMS 2000 subject classifications: Primary 60F17; secondary 62M10 Keywords and phrases: Fractional integration, Long memory, Nonlinear time series, Weak convergence
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Tse (1999) among others]. The FARIMA-GARCH model naturally fits into our framework and partially motivates us to study asymptotic properties of its partial sums. Most of the results in the literature assume {ut } to be either iid or linear processes. Recently, Wu and Min (2005) established an invariance principle under (1.2) when d ∈ [0, 1/2). The literature seems more concentrated on the case d ∈ (0, 1/2). Part of the reason is that this case corresponds to long memory and it appears in various areas such as finance, hydrology and telecommunication. When Pt Pt d ∈ (1/2, 1), the process is non-stationary and it can be defined as s=1 Xs or s=1 Ys , where Xs and Ys are Type I and Type II I(d − 1) processes, respectively. Empirical evidence of d ∈ (1/2, 1) has been found by Byers et al. (1997) in the poll data modeling and Kim (2000) in macroeconomics time series. Therefore the study of partial sums of I(d), d ∈ (−1/2, 0) is also of interest since it naturally leads to I(d) processes, d ∈ (1/2, 1). In fact, our results can be easily extended to the process with fractional index p + d, p ∈ N, d ∈ (−1/2, 0) ∪ (0, 1/2) [cf. Corollary 2.1]. The study of invariance principle has a long history. Here we only mention some representatives: Davydov (1970), Mcleich (1977), Gorodetskii (1977), Hall and Heyde (1980), Phillips and Solo (1992), Davidson and De Jong (2000), De Jong and Davidson (2000) and the references cited therein. Most of them deal with Type I processes. Recent developments for Type II processes can be found in Marinucci and Robinson (1999a), Wang et al. (2002) and Hosoya (2005) among others. The paper is structured as follows. Section 2 presents invariance principles for both types of processes. Section 3 considers limit distributions of tests of long memory under mild moment conditions. Technical details are given in the appendix. 2. Main Results We first define two types of fractional Brownian motions. For Type I fractional Brownian motion, let d ∈ (−1/2, 1/2) and Z ∞ 1 IBd (t) = [{(t − s)+ }d − {(−s)+ }d ]dIB(s), t ∈ R, A(d) −∞ where (t)+ = max(t, 0), IB(s) is a standard Brownian motion and A(d) =
1 + 2d + 1
Z
∞
[(1 + s)d − sd ]2 ds
1/2 .
0
Type II fractional Brownian motion {Wd (t), t ≥ 0}, d > −1/2, is defined as Wd (0) = 0, Wd (t) = (2d + 1)1/2
Z
t
(t − s)d dIB(s).
0
The main difference of IBd (t) and Wd (t) lies in the prehistoric treatment. See Marinucci and Robinson (1999b) for a detailed discussion of the difference between them. Pm Here we are interested in the weak convergence of the partial sums T = m i=1 Xi Pm and Tm = i=1 Yi . Let D[0, 1] be the space of functions on [0, 1] which are right continuous and have left-hand limits, endowed with the Skorohod topology (Billingsley, 1968). Denote weak convergence by ”⇒”. For a random variable X, write X ∈ Lp (p > 0) if kXkp := [E(|X|p )]1/p < ∞ and k · k = k · k2 . Let Ft = (· · · , εt−1 , εt ) be the shift process. Define the projections imsart-lnms ver.
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Pk by Pk X = E(X|Pk ) − E(X|Pk−1 ), X ∈ L1 . For two sequences (an ), (bn ), denote by an ∼ bn if an /bn → 1 as n → ∞. The symbols “→D ” and “→P ” stand for convergence in distribution and in probability, respectively. The symbols OP (1) and oP (1) signify being bounded in probability and convergence to zero in probability. Let N (µ, σ 2 ) be a normal distribution with mean µ and variance σ 2 . Hereafter we assume without loss of generality that E(ut ) = 0, µ = 0 and Y0 = 0. Let {ε0k , k ∈ Z} be an iid copy of {εk , k ∈ Z} and Fk∗ = (F−1 , ε00 , ε1 , . . . , εk ). Theorems 2.1 and 2.2 concern Type I and II processes respectively. Using the continuous mapping theorem, Theorems 2.1 and 2.2 imply Corollary 2.1 which deals with general fractional processes with higher orders. For d ∈ (−1/2, 0), an undesirable feature of our results is that the moment condition depends on d. However, this seems to be necessary; see Remark 4.1. Similar conditions were imposed in Sowell (1990) and Wang et al. (2003). Theorem 2.2 extends early results by Akonom and Gourieroux (1987), Tanaka (1999) and Wang et al. (2002), who assumed ut to be either iid or linear processes. See Marinucci and Robinson (1999a) and Hosoya (2005) for a multivariate extension. Theorem 2.1. Let ut ∈ Lq , where q > 2/(2d + 1) if d ∈ (−1/2, 0) and q = 2 if d ∈ (0, 1/2). Assume ∞ X
kP0 uk kq < ∞.
(2.1)
k=0
Then
P∞
k=j
Pj uk → ζj in Lq and, if kζ0 k > 0,
Tbntc A(d)kζ0 k . ⇒ κ1 (d)IBd (t) in D[0, 1], where κ1 (d) = Γ(d + 1) nd+1/2
(2.2)
Remark 2.1. Note that kζ0 k2 = 2πfu (0), where fu (·) is the spectral density function of {ut }; see Wu (2005b) and Wu and Min (2005) for the details. Theorem 2.2. Under the conditions of Theorem 2.1, for type II processes, we have Tbntc
nd+1/2
⇒ κ2 (d)Wd (t) in D[0, 1], where κ2 (d) =
kζ0 k(2d + 1)−1/2 . Γ(d + 1)
(2.3)
Corollary 2.1. Let ut satisfy conditions in Theorem 2.1; let d ∈ (0, 1/2) ∪ ˜ t by (1 − B)p+d X ˜ t = ut . Then (−1/2, 0) and p ∈ N. [a] Define the process X −(d+p−1/2) ˜ Xbntc ⇒ κ1 (d)IBd,p (t) in D[0, 1]; (i). n Pbntc ˜ −(d+p+1/2) (ii). n j=1 Xj ⇒ κ1 (d)IBd,p+1 (t) in D[0, 1]; Rt Pbntc ˜ 2 2 −2(d+p) 2 (iii). n j=1 Xj ⇒ κ1 (d) 0 [IBd,p (s)] ds in D[0, 1]. Here IBd,p (t) is defined as IBd,p (t) =
R t R tp−1 0
0
···
R t2 0
IBd (t), IBd (t1 )dt1 dt2 · · · dtp−1 ,
p = 1, p ≥ 2.
[b] Define the process Y˜t by (1 − B)p+d Y˜t = ut 1(t ≥ 1). Then similarly (i). n−(d+p−1/2) Y˜bntc ⇒ κ2 (d)Wd+p−1 (t) in D[0, 1]; Pbntc (ii). n−(d+p+1/2) j=1 Y˜j ⇒ κ2 (d)Wd+p (t) in D[0, 1]; Rt Pbntc ˜ 2 (iii). n−2(d+p) Y ⇒ κ2 (d) [Wd+p−1 (s)]2 ds in D[0, 1]. j=1
j
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We now discuss condition (2.1). Let gk (F0 ) = E(uk |F0 ) and δq (k) = kgk (F0 ) − gk (F0∗ )kq . Then δq (k) measures the contribution of ε0 in predicting uk . In Wu (2005a) it is called the predictive dependence measure. Since kP0 uk kq ≤ δq (k) ≤ 2kP0 uk kq (Wu, 2005a), (2.1) is equivalent to the q-stability condition (Wu, 2005a) P ∞ k=0 δq (k) < ∞, which suggests short-range dependence in that the cumulative contribution of ε0 in predicting future values of uk is finite. For a variety of nonlinear time series, δq (k) = O(ρk ) for some ρ ∈ (0, 1). The latter is equivalent to the geometric moment contraction (GMC) [Wu and Shao (2004), Wu and Min (2005)]. Shao and Wu (2005) verified GMC for GARCH(r, s) model and its asymmetric variants and showed P that the GMC property is preserved under ARMA filter. In P∞ ∞ the special case ut = k=0 bk εt−k , (2.1) holds if k=0 |bk | < ∞ and ε1 ∈ Lq . 3. Applications There have been a large amount of work on test of long memory under short memory null hypothesis, i.e. I(0) versus I(d), d ∈ (0, 1/2). For example, Lo (1991) introduced modified R/S test statistics, which admits the following form: k k X X 1 ¯n) , ¯ n ) − min (Xj − X (Xj − X max Qn = 1≤k≤n wn,l 1≤k≤n j=1
j=1
Pn −1
¯n = n where X j=1 Xj is the sample mean and wn,l is the long run variance estimator of Xt . Following Lo (1991), 2 wn,l
l n X 1X j 2 ¯ = 1− (Xj − Xn ) + 2 γˆj , n j=1 l+1 j=1
(3.1)
Pn−j ¯ n )(Xi+j − X ¯ n ), 0 ≤ j < n. The form (3.1) is equivalent where γˆj = n1 i=1 (Xi − X to the nonparametric spectral density estimator of {Xt } evaluated at zero frequency with Bartlett window (up to a constant factor). Here the bandwidth satisfies l = l(n) → ∞ and l/n → 0, as n → ∞.
(3.2)
Lee and Schmidt (1996) applied the KPSS test (Kwiatkowski et al. 1992) for I(0) versus I(d), d ∈ (−1/2, 0) ∪ (0, 1/2). The test statistics has the form: 2 k n X X 1 ¯ n ) (Xj − X Kn = 2 2 wn,l n j=1 k=1
2 with wn,l given by (3.1). Lee and Schmidt showed that the test is consistent against fractional alternatives and derived its asymptotic distribution under the assumption that ut are iid normal random variables. Giraitis et al. (2003) investigated the theoretical performance of various forms of nonparametric tests under both short memory hypotheses and long memory alternatives. In a quite general setting, we obtain asymptotic distributions of R/S and KPSS test statistics under fractional alternatives.
Theorem 3.1. Suppose that Xt is generated from (1.2) and ut satisfies (2.1) with some q > max(2, 2/(2d + 1)). Assume (3.2). Then 2 l−2d wn,l →P κ21 (d).
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Consequently, we have ld nd+1/2
˜ d (t) − inf IB ˜ d (t), Qn →D sup IB 0≤t≤1
0≤t≤1
(3.4)
˜ d (t) is the fractional Brownian bridge, i.e. IB ˜ d (t) = IBd (t) − tIBd (1), and where IB l2d Kn →D n2d
Z
1
˜d (t))2 dt. (B
(3.5)
0
Remark 3.1. For d ∈ (0, 1/2), Giraitis et al. (2003) obtained (3.4) under the joint cumulant summability condition sup
n X
|cum(X0 , Xh , Xr , Xs )| = O(n2d ).
(3.6)
h r,s=−n
For linear processes, (3.6) can be verified. But for nonlinear fractional processes (1.2), it seems hard to directly verify (3.6). In contrast, we only need to impose q-th (q > 2) moment condition when d ∈ (0, 1/2). Our dependence condition (2.1) can be easily verified for various nonlinear time series models [cf. Wu and Min (2005) and Shao and Wu (2005)]. 4. Appendix Lemma 4.1. Let ai = i−β `(i), i ≥ 1, where ` is a slowly varying function and β > 1/2; let q > (3/2 − β)−1 if 1 < β < 3/2 and q = 2 if 1/2 < β < 1; let σn = A(1 − β)n3/2−β `(n)/(1 P − β) and σ ˜n = (3 − 2β)−1/2 n3/2−β `(n)/(1 − β). ∞ ◦ Assume either (1 ) 1 < β < 3/2, i=0 ai = 0 or (2◦ ) 1/2 < β < 1. Further assume that the martingale differences ζj = F (. . . , εj−1 , εj ), j ∈ Z,
(4.1)
P∞ Pj−1 Pi satisfy ζj ∈ Lq . Let Yj = i=0 ai ζj−i , Yj = i=0 ai ζj−i , Si = j=1 Yj , Si = Pi −1 ⇒ ˜n−1 Sbntc j=1 Yj . Then we have [a] σn Sbntc ⇒ kζ0 kIB3/2−β (t) in D[0, 1] and [b] σ kζ0 kW3/2−β (t) in D[0, 1]. Proof of Lemma 4.1. [a] Consider (1◦ ) first. For the finite dimensional convergence, we shall apply the Cramer-Wold device. Fix 0 ≤ t1 < t2 ≤ 1 and let m1 = bnt1 c Pi and m2 = bnt2 c. Let Ai = j=0 aj if i ≥ 0 and Ai = 0 if i < 0. For λ, µ ∈ R let λ(Am1 −m2 +l − Al−m2 ) + µ(Al − Al−m2 ) , σn 2 σλµ = kζ0 k2 [λ2 t3−2β + µ2 t3−2β + λµ(t3−2β + t3−2β − (t2 − t1 )3−2β )]. 1 2 1 2 P∞ Then (λSm1 + µSm2 )/σn = l=0 cn,l ζm2 −l has martingale difference summands and we can apply the martingale central limit theorem. By Karamata’s Theorem, P∞ An = − j=n+1 aj ∼ n1−β `(n)/(β − 1). Elementary calculations show that cn,l
=
∞ X
2 c2n,l → σλµ and sup |cn,l | → 0 as n → ∞.
l=0
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2 Let Vl = E(ζm |Fm2 −l−1P ). By the argument in the proof of Theorem 1 in Hannan 2 −l ∞ 2 (1973), (4.2) implies that l=0 c2n,l Vl → σλµ in L1 . For completeness we prove it 0 00 here. Let ω > 0 be fixed, Vl = Vl 1Vl ≤ω and Vl = Vl − Vl0 . By (4.2),
2 ∞ ∞
X
X
0 c2n,l c2n,l0 |cov(V00 , Vl−l c2n,l (Vl0 − EVl0 ) ≤ lim sup lim sup 0 )| = 0
n→∞ n→∞ 0 l,l =0
l=0
since
limk→∞ cov(V00 , Vk0 )
= 0. Therefore, using Vl = Vl0 + Vl00 , again by (4.2), ∞ ∞ X X 2 2 00 00 cn,l (Vl − EVl ) lim sup E cn,l (Vl − EVl ) ≤ lim sup E n→∞ n→∞ l=0 l=0 ∞ X ≤ 2 lim sup c2n,l EVl00 → 0 as ω → ∞. n→∞
l=0
P∞
2 2 l=0 E{|cn,l ζm2 −l |1|cn,l ζm2 −l |≥δ }
Under (4.2), for any δ > 0, → 0. So the finite dimensional convergence holds. By Proposition 4 of Dedecker and Doukhan (2003), kSn k2q ≤ 2qkζ0 k2q
∞ X
(Aj − Aj−n )2 = O(σn2 ).
j=1
By Theorem 2.1 of Taqqu (1975), the tightness follows. (2◦ ) Note that kSn k ∼ kζ0 kσn , the conclusion similarly follows. [b] The finite dimensional convergence follows in the same manner as [a]. For the tightness, let 1 ≤ m1 < m2 ≤ n, by Proposition 4 in Dedecker and Doukhan (2003), kSm − Sm k2 ≤ 2qkζ0 k2q 2 1 q
m 2 −1 X
2 (Aj − Aj−(m2 −m1 ) )2 = O(˜ σm ). 2 −m1
j=0
With the above inequality, using the same argument as in Theorem 2.1 of Taqqu (1975), we have for any 0 ≤ t1 ≤ t ≤ t2 ≤ 1, there exists a generic constant C (independent of n, t1 , t, t2 ), such that for β ∈ (1/2, 1), E|Sbntc − Sbnt ||Sbnt − Sbntc | ≤ Cσ ˜n2 (t2 − t1 )3−2β 1c 2c
and for β ∈ (1, 3/2), E|Sbntc − Sbnt |q/2 |Sbnt − Sbntc |q/2 ≤ C σ ˜nq (t2 − t1 )q(3/2−β) . 1c 2c
Thus the tightness follows from Theorem 15.6 in Billingsley (1968).
♦
Remark 4.1. Under (1◦ ) of Lemma 4.1, the moment condition ζj ∈ Lq , q > (3/2 − β)−1 , is optimal. and it can not be reduced to ζj ∈ Lq0 , q0 = (3/2 − β)−1 . Consider the case in which ζi are iid symmetric random variables and P(|ζ0 |q0 ≥ g) ∼ g −1 (log g)−2 as g → ∞. Then ζj ∈ Lq0 . Let `(n) = 1/ log n, n > 3. Elementary calculations show that σn−1 max1≤j≤n |ζj | → ∞ in probability. Let Pi 0 0 0 Yj0 = Yj + ζj − ζj−1 and Si0 = j=1 Yj . Then the coefficients aj of Yj also 0 , satisfy the conditions in Lemma 4.1. The two processes σn−1 Sbntc and σn−1 Sbntc 0 ≤ t ≤ 1, cannot both converge weakly to fractional Brownian motions. If so, since maxj≤n |ζj − ζ0 | ≤ maxj≤n |Sj | + maxj≤n |Sj0 |, we have maxj≤n |ζj | = OP (σn ), contradicting σn−1 max1≤j≤n |ζj | → ∞ in probability. Similar examples are given in Wu and Min (2005) and Wu and Woodroofe (2004). ♦ imsart-lnms ver.
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Pk Proof of Theorem 2.1. Let aj = Γ(j + d)/{Γ(d)Γ(j + 1)}, j P ≥ 0, and Ak = i=0 ai ∞ if k ≥ 0 and 0 if k < 0. Note that β = 1 − d. Then Xt = j=0 aj ut−j . By (2.1), P∞ P P P∞ n n ζj = i=j Pj ui ∈ Lq . Let Mn = i=1 ζi , Sn = j=1 Yj , Yj = i=0 ai ζj−i , Un = P √ n i=1 ui and Rn = Tn − Sn . By Theorem 1 in Wu (2005b), kUn − Mn kq = o( n). By Karamata’s theorem and summation by parts, we have
3m
X
kRm kq ≤ (Ai − Ai−m )(um−i − dm−i )
i=0 q
∞
X
+ (Ai − Ai−m )(um−i − dm−i )
i=3m+1
=
3m X i=1
+
q
√ |(Ai − Ai−m ) − (Ai−1 − Ai−1−m )|o( i) ∞ X
√ |(Ai − Ai−m ) − (Ai−1 − Ai−1−m )|o( i) = o(σm ).
i=3m+1
By Proposition 1 in Wu (2005b),
max |Rm |
m≤2k
q
≤
k X
2(k−r)/q kR2r kq =
r=0
k X
2(k−r)/q o(σ2r ) = o(σ2k ),
r=0
♦
since q > 2/(2d + 1). So the theorem follows from Lemma 4.1. Proof of Theorem 2.2. As in the proof of Theorem 2.1, let Si = . By Karamata’s theorem, − Sm = Tm Rm kRm+l − Rl kq ≤
m X
Pi
j=1
Yj and
p |Aj − Aj−1 |o( j) = o(˜ σm ).
j=1
Again by the maximal inequality [Proposition 1 in Wu (2005b)], k−r 1/q
2X k X
max |Rm kR2r j − R2r (j−1) kqq | ≤
m≤2k
q
r=0
=
k X
j=1
2(k−r)/q o(˜ σ2r ) = o(˜ σ2k ),
r=0
which proves the theorem in view of Lemma 4.1.
♦
Proof of Theorem 3.1. If (3.3) holds, by the continuous mapping theorem, Theorem 2.1 entails (3.4) and (3.5). In the sequel we shall prove (3.3). Note that 2 wn,l
=
n−j n l X 1X 2 2X j ¯ n2 Xj + 1− Xi Xi+j − X n j=1 n j=1 l + 1 i=1 n−j l l X ¯n X ¯ n2 X 2X j 2X j − 1− (Xi + Xi+j ) + 1− (n − j) n j=1 l + 1 i=1 n j=1 l+1
¯ n2 + I2n . =: I1n − X imsart-lnms ver.
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8
¯ n | = OP (nd−1/2 ), l−2d (X ¯ n2 + |I2n |) = OP ((l/n)1−2d ) = oP (1). Thus it Since |X Pn Pj suffices to show that l−2d I1n →P κ21 (d). Let Vj = i=1 Xi , V˜j = i=n−j+1 Xi , 1 ≤ j ≤ l, then a straightforward calculation shows that I1n = J1n + J2n , where
J1n :=
1 (l + 1)n
n X
i X
i=l+1
2
l
Xj , J2n :=
j=i−l
X 1 (V 2 + V˜j2 ). n(l + 1) j=1 j
oP (l2d ). Corollary 2.1 implies that J2n = OP (l2d+2 /(ln)) =P Pn ∞ −2d 2 It remains to show l J1n →P κ1 (d). Let ζj = i=j Pj ui ∈ Lq ; Mn = i=1 ζi , Pn √ Un = i=1 ui and rn =√supj≥n kUj − Mj k/ j. Then rn → 0 and it is nonincreasing. Let L = bmin{ nl, l(r√l )1/(2β−3) }c. Then l = o(L) and L = o(n). Let Wj,l =
L X
(Ai − Ai−l )uj−i ,
Qj,l =
i=0
L X
(Ai − Ai−l )ζj−i
i=0
and b = bn/(2L)c. Since τm := E|E(ζ02 |F−m ) − E(ζ02 )| → 0 as m → ∞, E|E(Q22Lk,l |F2Lk−2L ) − E(Q22Lk,l )|
≤
L X
(Ai − Ai−l )2 τ2L−i = o(σl2 ). (4.3)
i=0
Let Dk = Q22Lk,l − E(Q22Lk,l |F2Lk−2L ), k = 1, . . . , b. Set Cq = 18q 3/2 (q − 1)−1/2 and q 0 = min(q, 4). By Burkholder’s inequality,
b
X
0 0
Dk ≤ Cq/2 b2/q kDk kq/2 ≤ 2Cq/2 b2/q kQ2Lk,l k2q
k=1
q/2
0
= b2/q O(σl2 ) = o(bσl2 ).
(4.4)
By (4.3) and (4.4), it is easily seen that n X 2 E {Qj,l − E(Q2j,l )} = o(nσl2 ). j=1 Let Sj,l =
Pj
i=j−l+1
(4.5)
Xi . Since l/L → 0,
∞
X
kSj,l − Wj,l k = (Ai − Ai−l )uj−i
i=L+1 " ∞ #1/2 ∞ X X ≤ (Ai − Ai−l )2 kP0 ut k = o(σl ).
(4.6)
t=0
i=L+1
By the definition of L, using summations by parts, we have kWj,l − Qj,l k
≤
L X
√ |(Ai − Ai−l ) − (Ai−1 − Ai−l−1 )|ri i
i=1
≤ r√ l
imsart-lnms ver.
L X √ i=1+ l
√
l X √ √ 2|ai |ri i = o(σl ). 2|ai | i +
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i=1
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9
By (4.6) and (4.7), kSj,l − Qj,l k = o(σl ). Hence by (4.5), X X n n 2 2 2 E|Sj,l E {Sj,l − E(Qj,l )} ≤ − Q2j,l | + o(nσl2 ) = o(nσl2 ) j=1 j=1 which completes the proof since E(Q2j,l ) ∼ σl2 kζ0 k2 .
♦
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