Stability of block-triangular stationary random ... - Semantic Scholar
Systems & Control Letters 57 (2008) 620–625 www.elsevier.com/locate/sysconle
Stability of block-triangular stationary random matrices L´aszl´o Gerencs´er a,∗ , Gy¨orgy Michaletzky a,b , Zsanett Orlovits a,c a MTA SZTAKI, Kendeu. 13-17, 1111 Budapest, Hungary b Department of Probability Theory and Statistics, E¨otv¨os Lor´and University, (ELTE) P´azm´any P´eter s´et´any 1/C, 1117 Budapest, Hungary c Department of Differential Equations, Budapest University of Technology and Economics (BME), M˝uegyetem rkp. 3-9., 1111 Budapest, Hungary
Received 11 April 2006; received in revised form 9 November 2007; accepted 7 January 2008 Available online 5 March 2008
Abstract The objective of this note is to prove, under certain technical conditions, that the top-Lyapunov exponent of a strictly stationary random sequence of block-triangular matrices is equal to the maximum of the top-Lyapunov exponents of its diagonal blocks. This study is partially motivated by a basic technical problem in the identification of GARCH processes. A recent extension of the above inheritance theorem in the context of L q -stability will also be briefly described. c 2008 Elsevier B.V. All rights reserved.
Keywords: Random coefficient linear stochastic systems; Identification; Lyapunov exponents; Random matrix products; GARCH processes
1. Introduction and the main result Consider a linear stochastic system given by the state-space equation of the form X n+1 = An+1 X n + u n+1 ,
−∞ < n < +∞,
(1)
where X n ∈ and (An , u n ) is a jointly strictly stationary sequence of random matrices of size p × ( p + 1) over some probability space (Ω , F, P), where Ω is the space or set of elementary events denoted by ω, the σ -algebra F is the set of measurable subsets of Ω , and P is a probability measure on F. A strictly stationary solution (X n ) is called causal if X n is measurable with respect to the σ -field σ {Ai , u i , i ≤ n}. Both necessary and sufficient conditions for the existence of a strictly stationary causal solution of (1) have been given in [6]. The proof of necessity is far from simple. To formulate a sufficient condition we need the concept of a Lyapunov exponent. Let A = (An ) be a strictly stationary, ergodic sequence of p × p random matrices over (Ω , F, P) such that Rp,
E log+ kAn k < +∞,
(2)
∗ Corresponding address: MTA SZTAKI, Computer and Automation
Institute of the Hungarian Academy of Sciences, 13-17 Kende utca, 1111 Budapest, Hungary. Tel.: +36 1 279 6138; fax: +36 1 4667 503. E-mail addresses: [email protected] (L. Gerencs´er), [email protected] (G. Michaletzky), [email protected] (Z. Orlovits). c 2008 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2008.01.001
where log+ x denotes the positive part of log x. Then it easy to see that 1 λ = lim E log kAn . . . A1 k n→∞ n exists, where −∞ ≤ λ < +∞. The number λ is called the topLyapunov exponent of A, and is denoted by λ(A). If An = A for all n then λ(A) is simply the logarithm of the spectral radius of A. A major result of the theory of random matrices is the F¨urstenberg–Kesten theorem stating that 1 log kAn . . . A1 k n almost surely, see [11]. It follows that for all ε > 0 the random variable λ(A) = lim
n→∞
c A (ω) = sup (log kAn . . . A1 k − n(λ + ε)) n≥0
exists and is finite almost surely, and thus kAn . . . A1 k ≤ C A (ω)en(λ+ε) ,
(3)
with C A (ω) = ec A (ω) . In the case when the range of (An ) is bounded, a simple crude upper bound of λ(A) is the so-called joint spectral radius of A. For its computation a sequence of useful results has been given in [2,4].
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Now assume that (An , u n ) is a strictly stationary, ergodic sequence of p × ( p + 1) random matrices such that (2) holds. Assume in addition that a similar condition holds for (u n ), i.e. E log+ ku n k < +∞.
(4)
Then it is not difficult to show, see [6], that if the topLyapunov exponent λ(A) is negative then (1) has a unique strictly stationary, causal solution given by X n∗ = u n +
∞ X
An An−1 . . . An−k+1 u n−k .
(5)
Furthermore, solving (1) with any initial condition X 0 forward in time we get that for any ε > 0 we have with probability 1 (w.p.1) X n − X n∗ = O(e(λ+ε)n ).
(6)
A nice application of linear stochastic systems with random system matrices is in the analysis of the celebrated GARCH model, which has been developed to describe stochastic volatility processes. It is well known for its capability to capture the phenomenon called volatility clustering; see [9,5]. Letting (yn ) denote the log-return of a stock on day n, it is assumed that (7)
n ∈ Z,
where (εn ) is a sequence of i.i.d. random variables with Eε0 = 0 and Eε02 = 1, and σn2 is defined via the feedback σn2 = α0∗ +
p X
2 αi∗ yn−i +
i=1
q X
2 β ∗j σn− j,
be a stationary, ergodic sequence of ( p +q)×( p +q) matrices, satisfying (2), with A1n and A2n being square matrices. Then λ(A) = max(λ(A1 ), λ(A2 )).
1 X n1 = A1n X n−1 + u 1n
β ∗j
X n+1 = An+1 X n + u n+1 , X θ,n+1 = Aθ,n+1 X n + An+1 X θ,n + u θ,n+1 ,
Theorem 1.1. Let 1 An 0 An = Bn A2n
Now consider a linear stochastic system
j=1
αi∗ ,
The proof of [15] follows a route completely different from what will be developed in the present paper, and apparently cannot be generalized to cover the general case to be discussed here. The main result of the present paper is a positive answer to this problem in a general setting under significantly weaker conditions.
(8)
> 0 and where ≥ 0. Now (7) and (8) can be written in a state-space form with 2 2 2 T state vector X n = (yn2 , . . . , yn− p+1 , σn , . . . , σn−q+1 ) . It is easy to see that (X n ) satisfies a linear state-space equation of the form (1) with (An , u n ) being an i.i.d sequence satisfying (2), such that their elements depend linearly on the system parameters. Another application of random matrices in finance is given in [14]. Now assume that the matrices (An , u n ) depend on a parameter θ ∈ D ⊂ Rk , where D is an open set, and assume that (An (θ ), u n (θ )) are C 1 -functions of θ for all ω ∈ Ω . Assume that the conditions of [6] described above are satisfied. Then the unique strictly stationary, causal solution of (1) will be denoted by (X n (θ )). In system identification we typically need to differentiate a parameter-dependent state-vector. It is not obvious at all that X n (θ ) is differentiable for all ω or for almost all ω ∈ Ω . Let X θ,n denote the derivative of X n (θ ) with respect to the parameter vector θ. Carrying out formal differentiation and assuming that θ is scalar we get for the extended state vector (X n , X θ,n ) the state-space equation α0∗
¯ < 0, and if an We may then ask if λ(A) < 0 implies λ(A) approximation similar to (6) is possible for the solutions of the extended system (9). A positive answer to this problem has been given in Mikosch and Straumann [15] under the condition that the sequence (An ) is independent and identically distributed (i.i.d.), and for some s > 0 Ek A¯ 1 ks < +∞.
k=1
yn = σn εn ,
transition matrix will be An 0 A¯ n = . Aθ,n An
(9) (10)
where Aθ,n+1 and u θ,n+1 denotes the derivatives of An+1 (θ ) and u n+1 (θ ) with respect to θ, respectively. Thus the state
X n2
=
2 A2n X n−1
+
1 Bn X n−1
(11) + u 2n ,
(12)
where X n = (X n1 , X n2 ) ∈ R p+q , u n = (u 1n , u 2n ), and (An , u n ) is a jointly strictly stationary sequence. Assume that u n satisfies condition (2). Then solving (11) with any initial condition X 0 forward in time we get that for any ε > 0 X n − X n∗ = O(e(λ+ε)n )
w.p.1.
(13)
An application and motivating example for the above theorem consider the problem of identifying the system parameters of a GARCH process. We proceed in a standard manner: for a fixed tentative value of the system parameters, say θ = (α0 , α1 , . . . , α p , β1 , . . . , βq )T we invert the system, using zero initial conditions, to get the assumed values σn2 (θ ) and εn (θ). Then the (conditional) quasi-log-likelihood function, modulo a constant, is N X 1 y2 L N (θ ) = − log σn2 (θ) + 2 n . 2 σn (θ ) n=1 To compute its gradient we need to differentiate σn2 (θ ). Using a state-space representation we are thus lead to a situation described in general above. For a second application consider again the random linear stochastic system (1) satisfying λ(A) < 0. We have seen that
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condition (4) is sufficient for the existence of a stationary and causal solution. We give an example, using Theorem 1.1, which shows that this condition might be relaxed. Consider serially coupled input–output systems given by the equations, with −∞ < n < ∞, 1 X n1 = A1n X n−1 + u 1n
X n2
=
2 A2n X n−1
u 1n
where ∈ Defining
+
Rp.
(14)
1 Bn X n−1 ,
(15)
Note that there is no exogenous input in (15).
1 u 2n = Bn X n−1 ,
(16)
we can write (15) as 2 X n2 = A2n X n−1 + u 2n .
(17)
Note, however, that the validity of (4) for u 2n cannot be guaranteed. In spite of this (17) is well defined under the conditions of the following theorem, a direct consequence of Theorem 1.1. Corollary 1.1. Let (An , u 1n ) be a stationary, ergodic sequence, jointly satisfying (2). Assume that λ(A1 ) < 0, λ(A2 ) < 0. Let (X n1 ) be the unique strictly stationary, causal solution of (14). Then the linear stochastic system (17) has a strictly stationary, causal solution given by 1 X n2 = Bn X n−1 +
∞ X
A2n A2n−1 . . . A2n−k+1 u 2n−k ,
(18)
k=1
where the right-hand side converges almost surely.
It can be easily seen that the inequality λ(A) max(λ(A1 ), λ(A2 )) holds trivially. We will show that
holds as well. It is easy to see that it is sufficient to show that λ(A1 ) < 0,
where | · | denotes the Euclidean norm in R p . We may then ask under what condition will the infinite sum given in (5) converge in L q . This problem has been considered in [10] in a Markov chain setting. Assuming that (An ) and (u n ) are i.i.d. sequences, which are also independent of each other, they have shown that for an even q a necessary and sufficient condition ⊗q for L q -convergence in (5) is that ρ(E(A1 )) < 1. Here ρ(.) is the spectral norm, and A⊗q denotes the qth Kronecker power of A. A simpler proof for sufficiency under slightly weaker conditions has been given in [13]. Theorem 1.1 can be extended in the context of L q -stability as follows, see [13]: Theorem 1.2. Let A1 0 A= B A2
⊗q
⊗q
ρ[E(A⊗q )] = max{ρ[E(A1 )]; ρ[E(A2 )]}.
(19)
implies λ(A) < 0.
(20)
λ(cA) = λ(A) + log c. Let c be such that λ(cA1 ) < 0 and λ(cA2 ) < 0. Then by (20) it follows that λ(cA) < 0. In other words, with γ = − log c, we have λ(A1 ) < γ
λ(A2 ) < γ
implies λ(A) < γ ,
from which the claim follows. To prove (20) we use the following result which is part of Oseledec’s theorem, see [16, 17]. In the theorem below |x| stands for the Euclidean norm of a vector x ∈ R p . Theorem 2.1. Let A = (An ) be a stationary, ergodic sequence of p × p matrices satisfying (2). Then there are random subspaces S(ω) ⊂ R p of fixed dimension, say, s < p, such that S is a measurable function from (Ω , F) to the Grassmanian manifold of s-dimensional subspaces of R p , and there exists a null-set Ω0 ⊂ Ω such that for ω 6∈ Ω0 and any x ∈ R p , x 6∈ S(ω) we have 1 log |An . . . A1 x| = λ(A). n
Remark 2.1. The original form of Oseledec’s theorem has been stated for non-singular random matrices, see [16], but an extension to possibly singular sequences has been given in [17]. Although there seems to be a gap in the arguments of [17], the proof of the above partial result is rigorous. Corollary 2.1. Under the conditions of Theorem 2.1 we have λ(A) < 0 if and only if for Lebesgue almost all x ∈ R p we have lim sup n→∞
1 log |An . . . A1 x| < 0 n
w.p.1.
(21)
We note in passing that, obviously, the hard part of the above result is to show that (21) implies λ(A) < 0. To prove (20) we apply Corollary 2.1. Thus it is sufficient to show that for Lebesgue almost all x ∈ R p+q lim sup
be a random (n 1 + n 2 ) × (n 1 + n 2 ) matrix in L q (Ω , F, P), with A1 and A2 being square matrices. Then for any positive integer, even or odd, we have
λ(A2 ) < 0
Indeed, for any constant c > 0 setting cA = (c An ) obviously
lim
for some q ≥ 1,
≥
λ(A) ≤ max(λ(A1 ), λ(A2 ))
n→∞
Consider again the linear stochastic system given by Eq. (1). Now assume that E|u n |q < +∞
2. Proof of the main theorem
n→∞
1 log |An . . . A1 x| < 0 n
w.p.1.
Writing 1 xn1 = A1n xn−1
x01 = x 1
1 2 xn2 = Bn xn−1 + A2n xn−1
x02 = x 2
(22)
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and solving this recursion we get xn1 = A1n . . . A11 x01 xn2 = A2n . . . A21 x02 +
(23) n X
A2n . . . A2k+1 Bk A1k−1 . . . A11 x01 .
(24)
Lemma 2.3. Let (ξn ), n ≥ 1 be a strictly stationary, ergodic process with Eξn = 0. Then for all ε > 0 there exists a random constant c(ω) such that for all 0 ≤ k < n we have w.p.1 ξn + · · · + ξk+1 ≤ c(ω) + nε.
k=1
To estimate xn1 (3) is applicable. Thus we get 1 +ε)
|A1n . . . A11 x 1 | ≤ C 1A (ω)en(λ
|x01 |
In the case when Eξn 6= 0 we have the following result: (25)
with λ1 = λ(A1 ). The first term on the right-hand side of (24) is estimated similarly. To estimate the norm of the second term of (24) we first apply the triangle inequality, and then estimate the kth term. We first estimate kA1k−1 . . . A11 k using the inequality (3) with (k − 1) replacing n. To estimate kBk k we use the following simple well-known lemma:
Lemma 2.4. Let (ξn ), n ≥ 1 be a strictly stationary, ergodic process such that Eξn exists and Eξn < ∞. Then for all finite µ ≥ Eξn , and for all ε > 0 there exists a random constant c(ω), such that for all 0 ≤ k < n we have w.p.1 ξn + · · · + ξk+1 ≤ c(ω) + (n − k)µ + nε.
Lemma 2.1. Let (ηk ), k ≥ 1 be a sequence of identically distributed real-valued random variables such that E log+ |ηk | < +∞. Then for any ε > 0 there exists a finite random variable C(ω) such that for all k ≥ 1
Proof. If Eξn is finite, then, applying Lemma 2.3 for the random variables ξ n = ξn −Eξn , the claim follows for µ = Eξn , thus, by monotonicity, it also follows for larger µ’s. If Eξn = −∞, then first define ξ n = ξn ∨(−K ), where K is large enough to ensure that Eξ n ≤ µ. Then the first part is applicable to get
|ηk | ≤ C(ω)eεk .
ξ n + · · · + ξ k+1 ≤ c(ω) + (n − k)µ + nε
In other words (ηk ) is sub-exponential.
Since ξ n ≥ ξn , we get the claim.
Thus we get that for any ε > 0 there exists a finite random variable C B such that for all k ≥ 1 kBk k ≤ C B eεk . Therefore the norm of the second term in (24) can be estimated from above by C 1A C B
n X
kA2n . . . A2k+1 kek(λ
1 +2ε)
|x01 |,
(26)
k=1
w.p.1.
We continue the proof of Theorem 1.1 by estimating kA2n . . . A2k+1 k first in the scalar case, i.e., when A2n is a scalar, say |A2n | = an . Then obviously λ2 = λ(A2 ) = E log an . Lemma 2.5. Let (an ), n ≥ 1 be a non-negative, strictly stationary, ergodic process such that λ = E log ak exists, and λ < ∞. Then for all finite µ ≥ λ, and for any ε > 0 there exists a random variable C(ω), such that for all 0 ≤ k < n
and here λ1 < 0. To estimate kA2n . . . A2k+1 k we prove a few simple auxiliary results:
an . . . ak+1 ≤ C(ω)e(n−k)µ enε
Lemma 2.2. Let (ξn ), n ≥ 1 be a strictly stationary, ergodic process with Eξn = 0. Then for all ε > 0 there exists a random constant c(ω) such that w.p.1
A key point in the above statement is that C(ω) is independent of n.
ξ1 + · · · + ξn ≤ c(ω) + nε. Proof. The proposition is a direct consequence of the strong law of large numbers, known as Birkhoff’s ergodic theorem, for strictly stationary, ergodic processes. Similarly, reversing the time, we would get for a two-sided, strictly stationary, ergodic process that ξn + · · · + ξk+1 ≤ cn (ω) + (n − k)ε. Unfortunately, the constant cn (ω) depends on n. To get an upper bound in which the constant does not depend on n, write ξn + · · · + ξk+1 = (ξ1 + · · · + ξn ) − (ξ1 + · · · + ξk ) ≤ 2 sup (ξ1 + · · · + ξk ) ≤ 2(c(ω) + nε). 1≤k≤n
Repeating this argument with ε/2 replacing ε we get the following lemma:
w.p.1.
(27)
Proof. Writing ξk = log ak , applying Lemma 2.4, then exponentiating the resulting inequality, we get the claim. Using this lemma with λ = λ(A2 ) = λ2 in (26), substituting the resulting upper bound into (24), and using the arguments following (24), we get after some simplifications that for µ = max(λ1 , λ2 ) < 0, and for any ε there exists a random variable C 0 (ω), such that for all n ≥ 1 we have |xn2 | ≤ C 0 (ω)en(µ+ε) , and thus (22) follows. To estimate kA2n . . . A2k+1 k in the general case we need a simple observation that states that the subadditive process log kA2n . . . A2k+1 k can be majorated by a scalar-valued additive process modulo negligible error with growth rate arbitrarily close to λ2 = λ(A2 ) < 0.
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Lemma 2.6. Let A = (An ) be a strictly stationary, ergodic sequence of p × p random matrices such that E log+ kAn k < +∞. Then for any ε > 0 there exists a scalar-valued, stationary and ergodic process (ξn ), and a finite random variable c(ω), such that Eξn < λ(A) + ε, and for any 0 ≤ k < n log kAn . . . Ak+1 k ≤ c(ω) + ξn + . . . + ξk+1 + nε with probability 1. It follows that, with ak = eξk we have kAn . . . Ak+1 k ≤ C(ω)an . . . ak+1 enε , and here E log an < λ(A) + ε. The proof is essentially given in [11], and will be restyled and given in the Appendix. Applying the above result for A = A2 = (A2n ) in (24), the proof of Theorem 1.1 can be completed as in the scalar case.
Remark 3.2. Consider Eq. (1), satisfying the conditions of Section 1, except λ(A) < 0. A possible route to establish λ(A) < 0 would be to establish the existence of a causal stationary solution directly, and then use the following deep result of [6]. Theorem 3.1. Consider the linear stochastic system given by the state-space equation (1), where X n ∈ R p , (An , u n ) is a jointly strictly stationary sequence of random matrices of size p × ( p + 1), jointly satisfying condition (2). Let us assume that the sequence (An , u n ) is controllable in the sense that there is no proper subspace V ⊂ R p , such that A0 V + u 0 ⊂ V
w.p.1.
Then if (1) has a strictly stationary causal solution (X n ), then λ(A) < 0. Unfortunately, the direct verification of the existence of a causal stationary solution does not seem to be easy.
3. Discussion In this section we present a few remarks to highlight the delicacy of the details of the proof of the main result. Remark 3.1. To estimate kAn . . . Ak+1 k, see Lemma 2.6, an alternative, direct approach would be to use the F¨urstenberg–Kesten theorem starting at time k, and using the estimate that for all fixed ε > 0 we have kAn . . . Ak+1 k ≤ Ck+1 (ω)e(n−k)(λ+ε) , where λ = λ(A). Recall that Ck+1 (ω) can be defined as Ck+1 (ω) = eck+1 (ω) , where ck+1 (ω) = sup (log kAn . . . Ak+1 k − (n − k)(λ + ε)). n≥k+1
Using a representation of (An ) via a measure-preserving shift on Ω it is easily seen that (Ck+1 ) can be assumed to be a stationary sequence. To control the effect of (Ck+1 ) in (24) we would need to show that (Ck+1 ) is sub-exponential. One way to show this would be to show that E log+ Ck+1 (ω) < +∞.
(28)
Unfortunately, this inequality is not true in general. Indeed, consider a scalar-valued i.i.d. process An = an . Then λ = E log an , and for fixed ε > 0 we have log Ck+1 (ω) = sup
n X
(log a j − (λ + ε)).
n≥k+1 j=k+1
According to the Kiefer–Wolfowitz theorem (see [7], Chapter 10.4, Corollary 3) (28) holds if and only if E(log+ a j )2 < +∞. The same remark applies if we estimate kAn . . . Ak+1 k using the F¨urstenberg–Kesten theorem backwards in time. Then we get for fixed n, for all fixed ε > 0, and for all k ≤ n kAn . . . Ak+1 k ≤ Cn (ω)e(n−k)(λ+ε) , and here the growth rate of Cn (ω) is, in general, not under control.
Remark 3.3. We note in passing, that an equally non-trivial deterministic version of the above result, with An taking its values from a given set of matrices A, has been given in [12]. For more on this subject see [1,8]. Remark 3.4. The existence of the derivative process (X θ,n ) in an almost sure sense, and, as a byproduct, the existence of a strictly stationary causal solution of (9) has been proved by direct arguments in [3] in the case of GARCH processes using specific arguments. Applying Theorem 3.1, we could conclude, with some additional work, that the top-Lyapunov exponent of the matrix An 0 ¯ An = . Aθ,n An is negative. It is not clear if this line of proof can used in the general case. Remark 3.5. For the sake of completeness we present the main tool that is used in the proof of [15]: Lemma 3.1. Let A = (An ) be an i.i.d. sequence of p × p matrices with EkAn ks < ∞ for some s > 0. Then for the associated top-Lyapunov exponent we have λ(A) < 0 if and only if there exist c > 0, s > 0 and 0 < ρ < 1 such that EkAn . . . A1 ks ≤ cρ n
for n ≥ 1.
The proof is based on the observation that for any fixed l ≥ 1 the map u 7→ EkAl . . . A1 ku has first derivative with respect to u equal to E log kAl . . . A1 k at u = 0. Acknowledgement The authors acknowledge the support of the National Research Foundation of Hungary (OTKA) under Grant no. T 047193.
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Appendix Proof of Lemma 2.6. We follow the proof of [11]. For a fixed ε > 0 take an l such that 1 E log kAl . . . A1 k < λ + ε. l For any 0 ≤ l < n and 0 ≤ r < l take a cover of the index set {k + 1, . . . , n} by l-tuples of the form Iqr = {ql + r, . . . , ql + r + l − 1},
By assumption 1l Eξi < λ + ε, and (ξn ) is strictly stationary and ergodic, hence, applying Lemma 2.4 we get that with some c(ω) depending on ε we have log kAn . . . Ak+1 k ≤
n 1 X ξi + (c00 (ω) + εn) l i=k+1
≤ c(ω) + (n − k)(λ + ε) + nε + (c00 (ω) + εn) which implies the claim.
a.s.
References
and let q0 = q0 (r ) = min{q : ql + r > k + 1} q1 = q1 (r ) = max{q : ql + r ≤ n}. Set k¯ = q0l + r and n¯ = q1l + r − 1. Then kAn . . . Ak+1 k ≤ kAn . . . An+1 ¯ k ×kAn¯ . . . An−l+1 k . . . kAk+l−1 . . . Ak¯ kkAk−1 . . . Ak+1 k, ¯ ¯ ¯ thus log kAn . . . Ak+1 k can be estimated from above by log kAn . . . An+1 ¯ k+
qX 1 −1
log kAql+r +l−1 . . . Aql+r k
q=q0
+ log kAk−1 . . . Ak+1 k. ¯
(29)
By Lemma 2.1 we have that for any ε0 > 0 there exists a random variable c0 (ω) such that log kAk k ≤ c0 (ω) + ε0 k. Thus, since n − (n¯ + 1) ≤ l we have log kAn . . . An+1 ¯ k ≤ log kAn k + · · · 0 0 + log kAn+1 ¯ k ≤ l(c (ω) + ε n).
A similar inequality holds for log kAk−1 . . . Ak+1 k. Now define ¯ ξi = log kAi+l−1 . . . Ai k ηqr = log kAql+r +l−1 . . . Aql+r k. Note that ηqr = ξql+r . Then the middle term in (29) can be written as qX 1 −1 q=q0
ηqr =
qX 1 −1
ξql+r .
q=q0
Letting r run from 0 to (l − 1) and averaging (29) over r we get ! X 1 n−l+1 ξi + 2l(c0 (ω) + ε 0 n) log kAn . . . Ak+1 k ≤ l i=k+1 ! n 1 X ξi + 3l(c0 (ω) + ε 0 (n + l)) a.s., ≤ l i=k+1 since ξi ≤ l(c0 (ω) + ε0 (n + l)) for any i ≤ n. For fixed l taking ε 0 sufficiently small we can upper bound the last, residual term as c00 (ω) + εn for any prescribed ε.
[1] N.E. Barabanov, On the Lyapunov exponent of discrete inclusions. I–III, Automation and Remote Control 49 (1988) 152–157, 283–287, 558–565. [2] N.E. Barabanov, Lyapunov exponent and joint spectral radius: Some known and new results, in: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, TuA08.1, 12–15 December, 2005, pp. 2332–2337. [3] I. Berkes, L. Horv´ath, P.S. Kokoszka, GARCH processes: Structure and estimation, Bernoulli 9 (2003) 201–217. [4] V.D. Blondel, Y. Nesterov, Computationally efficient approximations of the joint spectral radius, SIAM Journal on Matrix Analysis and Applications 27 (1) (2005) 256–272. [5] T. Bollerslev, Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics 31 (1986) 307–327. [6] P. Bougerol, N. Picard, Strict stationarity of generalized autoregressive processes, The Annals of Probability 20 (4) (1992) 1714–1730. [7] Y.S. Chow, H. Teicher, Independence, Interchangeability, Martingales, Springer Verlag, 1978. [8] W. Dayawansa, C.F. Martin, A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Transactions on Automatic Control 44 (1999) 751–760. [9] R.F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of the United Kingdom inflation, Econometrica 50 (1982) 987–1008. [10] P.D. Feigin, R.L. Tweedie, Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments, Journal of Time Series Analysis 6 (1) (1985) 1–14. [11] H. F¨urstenberg, H. Kesten, Products of random matrices, Annals of Mathematical Statistics 31 (1960) 457–469. [12] Gy. Michaletzky, L. Gerencs´er, BIBO stability of linear switching systems, IEEE Transactions on Automatic Control 47 (11) (2002) 1895–1898. [13] L. Gerencs´er, Zs. Orlovits, L q -stability of products of block-triangular stationary random matrices, Acta Scientiarum Mathematicarum (Szeged) (2007) (in press). [14] L. Gerencs´er, M. R´asonyi, Cs. Szepesv´ari, Zs. V´ag´o, Log-optimal currency portfolios and control Lyapunov exponents, in: Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC’05, Seville, Spain, MoC11.4, 12–15 December, 2005, pp. 1764–1769. [15] T. Mikosch, D. Straumann, Stable limit of martingale transforms with application to the estimation of GARCH parameters, Annals of Statistics 34 (1) (2006) 493–522. [16] V.I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Transactions of Moscow Mathematical Society 19 (1968) 197–231. [17] M.S. Ragunathan, A proof of Oseledec’s multiplicative ergodic theorem, Israel Journal of Mathematics 42 (1979) 356–362.
Stability of block-triangular stationary random matrices L´aszl´o Gerencs´er a,∗ , Gy¨orgy Michaletzky a,b , Zsanett Orlovits a,c a MTA SZTAKI, Kendeu. 13-17, 1111 Budapest, Hungary b Department of Probability Theory and Statistics, E¨otv¨os Lor´and University, (ELTE) P´azm´any P´eter s´et´any 1/C, 1117 Budapest, Hungary c Department of Differential Equations, Budapest University of Technology and Economics (BME), M˝uegyetem rkp. 3-9., 1111 Budapest, Hungary
Received 11 April 2006; received in revised form 9 November 2007; accepted 7 January 2008 Available online 5 March 2008
Abstract The objective of this note is to prove, under certain technical conditions, that the top-Lyapunov exponent of a strictly stationary random sequence of block-triangular matrices is equal to the maximum of the top-Lyapunov exponents of its diagonal blocks. This study is partially motivated by a basic technical problem in the identification of GARCH processes. A recent extension of the above inheritance theorem in the context of L q -stability will also be briefly described. c 2008 Elsevier B.V. All rights reserved.
Keywords: Random coefficient linear stochastic systems; Identification; Lyapunov exponents; Random matrix products; GARCH processes
1. Introduction and the main result Consider a linear stochastic system given by the state-space equation of the form X n+1 = An+1 X n + u n+1 ,
−∞ < n < +∞,
(1)
where X n ∈ and (An , u n ) is a jointly strictly stationary sequence of random matrices of size p × ( p + 1) over some probability space (Ω , F, P), where Ω is the space or set of elementary events denoted by ω, the σ -algebra F is the set of measurable subsets of Ω , and P is a probability measure on F. A strictly stationary solution (X n ) is called causal if X n is measurable with respect to the σ -field σ {Ai , u i , i ≤ n}. Both necessary and sufficient conditions for the existence of a strictly stationary causal solution of (1) have been given in [6]. The proof of necessity is far from simple. To formulate a sufficient condition we need the concept of a Lyapunov exponent. Let A = (An ) be a strictly stationary, ergodic sequence of p × p random matrices over (Ω , F, P) such that Rp,
E log+ kAn k < +∞,
(2)
∗ Corresponding address: MTA SZTAKI, Computer and Automation
Institute of the Hungarian Academy of Sciences, 13-17 Kende utca, 1111 Budapest, Hungary. Tel.: +36 1 279 6138; fax: +36 1 4667 503. E-mail addresses: [email protected] (L. Gerencs´er), [email protected] (G. Michaletzky), [email protected] (Z. Orlovits). c 2008 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2008.01.001
where log+ x denotes the positive part of log x. Then it easy to see that 1 λ = lim E log kAn . . . A1 k n→∞ n exists, where −∞ ≤ λ < +∞. The number λ is called the topLyapunov exponent of A, and is denoted by λ(A). If An = A for all n then λ(A) is simply the logarithm of the spectral radius of A. A major result of the theory of random matrices is the F¨urstenberg–Kesten theorem stating that 1 log kAn . . . A1 k n almost surely, see [11]. It follows that for all ε > 0 the random variable λ(A) = lim
n→∞
c A (ω) = sup (log kAn . . . A1 k − n(λ + ε)) n≥0
exists and is finite almost surely, and thus kAn . . . A1 k ≤ C A (ω)en(λ+ε) ,
(3)
with C A (ω) = ec A (ω) . In the case when the range of (An ) is bounded, a simple crude upper bound of λ(A) is the so-called joint spectral radius of A. For its computation a sequence of useful results has been given in [2,4].
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Now assume that (An , u n ) is a strictly stationary, ergodic sequence of p × ( p + 1) random matrices such that (2) holds. Assume in addition that a similar condition holds for (u n ), i.e. E log+ ku n k < +∞.
(4)
Then it is not difficult to show, see [6], that if the topLyapunov exponent λ(A) is negative then (1) has a unique strictly stationary, causal solution given by X n∗ = u n +
∞ X
An An−1 . . . An−k+1 u n−k .
(5)
Furthermore, solving (1) with any initial condition X 0 forward in time we get that for any ε > 0 we have with probability 1 (w.p.1) X n − X n∗ = O(e(λ+ε)n ).
(6)
A nice application of linear stochastic systems with random system matrices is in the analysis of the celebrated GARCH model, which has been developed to describe stochastic volatility processes. It is well known for its capability to capture the phenomenon called volatility clustering; see [9,5]. Letting (yn ) denote the log-return of a stock on day n, it is assumed that (7)
n ∈ Z,
where (εn ) is a sequence of i.i.d. random variables with Eε0 = 0 and Eε02 = 1, and σn2 is defined via the feedback σn2 = α0∗ +
p X
2 αi∗ yn−i +
i=1
q X
2 β ∗j σn− j,
be a stationary, ergodic sequence of ( p +q)×( p +q) matrices, satisfying (2), with A1n and A2n being square matrices. Then λ(A) = max(λ(A1 ), λ(A2 )).
1 X n1 = A1n X n−1 + u 1n
β ∗j
X n+1 = An+1 X n + u n+1 , X θ,n+1 = Aθ,n+1 X n + An+1 X θ,n + u θ,n+1 ,
Theorem 1.1. Let 1 An 0 An = Bn A2n
Now consider a linear stochastic system
j=1
αi∗ ,
The proof of [15] follows a route completely different from what will be developed in the present paper, and apparently cannot be generalized to cover the general case to be discussed here. The main result of the present paper is a positive answer to this problem in a general setting under significantly weaker conditions.
(8)
> 0 and where ≥ 0. Now (7) and (8) can be written in a state-space form with 2 2 2 T state vector X n = (yn2 , . . . , yn− p+1 , σn , . . . , σn−q+1 ) . It is easy to see that (X n ) satisfies a linear state-space equation of the form (1) with (An , u n ) being an i.i.d sequence satisfying (2), such that their elements depend linearly on the system parameters. Another application of random matrices in finance is given in [14]. Now assume that the matrices (An , u n ) depend on a parameter θ ∈ D ⊂ Rk , where D is an open set, and assume that (An (θ ), u n (θ )) are C 1 -functions of θ for all ω ∈ Ω . Assume that the conditions of [6] described above are satisfied. Then the unique strictly stationary, causal solution of (1) will be denoted by (X n (θ )). In system identification we typically need to differentiate a parameter-dependent state-vector. It is not obvious at all that X n (θ ) is differentiable for all ω or for almost all ω ∈ Ω . Let X θ,n denote the derivative of X n (θ ) with respect to the parameter vector θ. Carrying out formal differentiation and assuming that θ is scalar we get for the extended state vector (X n , X θ,n ) the state-space equation α0∗
¯ < 0, and if an We may then ask if λ(A) < 0 implies λ(A) approximation similar to (6) is possible for the solutions of the extended system (9). A positive answer to this problem has been given in Mikosch and Straumann [15] under the condition that the sequence (An ) is independent and identically distributed (i.i.d.), and for some s > 0 Ek A¯ 1 ks < +∞.
k=1
yn = σn εn ,
transition matrix will be An 0 A¯ n = . Aθ,n An
(9) (10)
where Aθ,n+1 and u θ,n+1 denotes the derivatives of An+1 (θ ) and u n+1 (θ ) with respect to θ, respectively. Thus the state
X n2
=
2 A2n X n−1
+
1 Bn X n−1
(11) + u 2n ,
(12)
where X n = (X n1 , X n2 ) ∈ R p+q , u n = (u 1n , u 2n ), and (An , u n ) is a jointly strictly stationary sequence. Assume that u n satisfies condition (2). Then solving (11) with any initial condition X 0 forward in time we get that for any ε > 0 X n − X n∗ = O(e(λ+ε)n )
w.p.1.
(13)
An application and motivating example for the above theorem consider the problem of identifying the system parameters of a GARCH process. We proceed in a standard manner: for a fixed tentative value of the system parameters, say θ = (α0 , α1 , . . . , α p , β1 , . . . , βq )T we invert the system, using zero initial conditions, to get the assumed values σn2 (θ ) and εn (θ). Then the (conditional) quasi-log-likelihood function, modulo a constant, is N X 1 y2 L N (θ ) = − log σn2 (θ) + 2 n . 2 σn (θ ) n=1 To compute its gradient we need to differentiate σn2 (θ ). Using a state-space representation we are thus lead to a situation described in general above. For a second application consider again the random linear stochastic system (1) satisfying λ(A) < 0. We have seen that
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condition (4) is sufficient for the existence of a stationary and causal solution. We give an example, using Theorem 1.1, which shows that this condition might be relaxed. Consider serially coupled input–output systems given by the equations, with −∞ < n < ∞, 1 X n1 = A1n X n−1 + u 1n
X n2
=
2 A2n X n−1
u 1n
where ∈ Defining
+
Rp.
(14)
1 Bn X n−1 ,
(15)
Note that there is no exogenous input in (15).
1 u 2n = Bn X n−1 ,
(16)
we can write (15) as 2 X n2 = A2n X n−1 + u 2n .
(17)
Note, however, that the validity of (4) for u 2n cannot be guaranteed. In spite of this (17) is well defined under the conditions of the following theorem, a direct consequence of Theorem 1.1. Corollary 1.1. Let (An , u 1n ) be a stationary, ergodic sequence, jointly satisfying (2). Assume that λ(A1 ) < 0, λ(A2 ) < 0. Let (X n1 ) be the unique strictly stationary, causal solution of (14). Then the linear stochastic system (17) has a strictly stationary, causal solution given by 1 X n2 = Bn X n−1 +
∞ X
A2n A2n−1 . . . A2n−k+1 u 2n−k ,
(18)
k=1
where the right-hand side converges almost surely.
It can be easily seen that the inequality λ(A) max(λ(A1 ), λ(A2 )) holds trivially. We will show that
holds as well. It is easy to see that it is sufficient to show that λ(A1 ) < 0,
where | · | denotes the Euclidean norm in R p . We may then ask under what condition will the infinite sum given in (5) converge in L q . This problem has been considered in [10] in a Markov chain setting. Assuming that (An ) and (u n ) are i.i.d. sequences, which are also independent of each other, they have shown that for an even q a necessary and sufficient condition ⊗q for L q -convergence in (5) is that ρ(E(A1 )) < 1. Here ρ(.) is the spectral norm, and A⊗q denotes the qth Kronecker power of A. A simpler proof for sufficiency under slightly weaker conditions has been given in [13]. Theorem 1.1 can be extended in the context of L q -stability as follows, see [13]: Theorem 1.2. Let A1 0 A= B A2
⊗q
⊗q
ρ[E(A⊗q )] = max{ρ[E(A1 )]; ρ[E(A2 )]}.
(19)
implies λ(A) < 0.
(20)
λ(cA) = λ(A) + log c. Let c be such that λ(cA1 ) < 0 and λ(cA2 ) < 0. Then by (20) it follows that λ(cA) < 0. In other words, with γ = − log c, we have λ(A1 ) < γ
λ(A2 ) < γ
implies λ(A) < γ ,
from which the claim follows. To prove (20) we use the following result which is part of Oseledec’s theorem, see [16, 17]. In the theorem below |x| stands for the Euclidean norm of a vector x ∈ R p . Theorem 2.1. Let A = (An ) be a stationary, ergodic sequence of p × p matrices satisfying (2). Then there are random subspaces S(ω) ⊂ R p of fixed dimension, say, s < p, such that S is a measurable function from (Ω , F) to the Grassmanian manifold of s-dimensional subspaces of R p , and there exists a null-set Ω0 ⊂ Ω such that for ω 6∈ Ω0 and any x ∈ R p , x 6∈ S(ω) we have 1 log |An . . . A1 x| = λ(A). n
Remark 2.1. The original form of Oseledec’s theorem has been stated for non-singular random matrices, see [16], but an extension to possibly singular sequences has been given in [17]. Although there seems to be a gap in the arguments of [17], the proof of the above partial result is rigorous. Corollary 2.1. Under the conditions of Theorem 2.1 we have λ(A) < 0 if and only if for Lebesgue almost all x ∈ R p we have lim sup n→∞
1 log |An . . . A1 x| < 0 n
w.p.1.
(21)
We note in passing that, obviously, the hard part of the above result is to show that (21) implies λ(A) < 0. To prove (20) we apply Corollary 2.1. Thus it is sufficient to show that for Lebesgue almost all x ∈ R p+q lim sup
be a random (n 1 + n 2 ) × (n 1 + n 2 ) matrix in L q (Ω , F, P), with A1 and A2 being square matrices. Then for any positive integer, even or odd, we have
λ(A2 ) < 0
Indeed, for any constant c > 0 setting cA = (c An ) obviously
lim
for some q ≥ 1,
≥
λ(A) ≤ max(λ(A1 ), λ(A2 ))
n→∞
Consider again the linear stochastic system given by Eq. (1). Now assume that E|u n |q < +∞
2. Proof of the main theorem
n→∞
1 log |An . . . A1 x| < 0 n
w.p.1.
Writing 1 xn1 = A1n xn−1
x01 = x 1
1 2 xn2 = Bn xn−1 + A2n xn−1
x02 = x 2
(22)
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and solving this recursion we get xn1 = A1n . . . A11 x01 xn2 = A2n . . . A21 x02 +
(23) n X
A2n . . . A2k+1 Bk A1k−1 . . . A11 x01 .
(24)
Lemma 2.3. Let (ξn ), n ≥ 1 be a strictly stationary, ergodic process with Eξn = 0. Then for all ε > 0 there exists a random constant c(ω) such that for all 0 ≤ k < n we have w.p.1 ξn + · · · + ξk+1 ≤ c(ω) + nε.
k=1
To estimate xn1 (3) is applicable. Thus we get 1 +ε)
|A1n . . . A11 x 1 | ≤ C 1A (ω)en(λ
|x01 |
In the case when Eξn 6= 0 we have the following result: (25)
with λ1 = λ(A1 ). The first term on the right-hand side of (24) is estimated similarly. To estimate the norm of the second term of (24) we first apply the triangle inequality, and then estimate the kth term. We first estimate kA1k−1 . . . A11 k using the inequality (3) with (k − 1) replacing n. To estimate kBk k we use the following simple well-known lemma:
Lemma 2.4. Let (ξn ), n ≥ 1 be a strictly stationary, ergodic process such that Eξn exists and Eξn < ∞. Then for all finite µ ≥ Eξn , and for all ε > 0 there exists a random constant c(ω), such that for all 0 ≤ k < n we have w.p.1 ξn + · · · + ξk+1 ≤ c(ω) + (n − k)µ + nε.
Lemma 2.1. Let (ηk ), k ≥ 1 be a sequence of identically distributed real-valued random variables such that E log+ |ηk | < +∞. Then for any ε > 0 there exists a finite random variable C(ω) such that for all k ≥ 1
Proof. If Eξn is finite, then, applying Lemma 2.3 for the random variables ξ n = ξn −Eξn , the claim follows for µ = Eξn , thus, by monotonicity, it also follows for larger µ’s. If Eξn = −∞, then first define ξ n = ξn ∨(−K ), where K is large enough to ensure that Eξ n ≤ µ. Then the first part is applicable to get
|ηk | ≤ C(ω)eεk .
ξ n + · · · + ξ k+1 ≤ c(ω) + (n − k)µ + nε
In other words (ηk ) is sub-exponential.
Since ξ n ≥ ξn , we get the claim.
Thus we get that for any ε > 0 there exists a finite random variable C B such that for all k ≥ 1 kBk k ≤ C B eεk . Therefore the norm of the second term in (24) can be estimated from above by C 1A C B
n X
kA2n . . . A2k+1 kek(λ
1 +2ε)
|x01 |,
(26)
k=1
w.p.1.
We continue the proof of Theorem 1.1 by estimating kA2n . . . A2k+1 k first in the scalar case, i.e., when A2n is a scalar, say |A2n | = an . Then obviously λ2 = λ(A2 ) = E log an . Lemma 2.5. Let (an ), n ≥ 1 be a non-negative, strictly stationary, ergodic process such that λ = E log ak exists, and λ < ∞. Then for all finite µ ≥ λ, and for any ε > 0 there exists a random variable C(ω), such that for all 0 ≤ k < n
and here λ1 < 0. To estimate kA2n . . . A2k+1 k we prove a few simple auxiliary results:
an . . . ak+1 ≤ C(ω)e(n−k)µ enε
Lemma 2.2. Let (ξn ), n ≥ 1 be a strictly stationary, ergodic process with Eξn = 0. Then for all ε > 0 there exists a random constant c(ω) such that w.p.1
A key point in the above statement is that C(ω) is independent of n.
ξ1 + · · · + ξn ≤ c(ω) + nε. Proof. The proposition is a direct consequence of the strong law of large numbers, known as Birkhoff’s ergodic theorem, for strictly stationary, ergodic processes. Similarly, reversing the time, we would get for a two-sided, strictly stationary, ergodic process that ξn + · · · + ξk+1 ≤ cn (ω) + (n − k)ε. Unfortunately, the constant cn (ω) depends on n. To get an upper bound in which the constant does not depend on n, write ξn + · · · + ξk+1 = (ξ1 + · · · + ξn ) − (ξ1 + · · · + ξk ) ≤ 2 sup (ξ1 + · · · + ξk ) ≤ 2(c(ω) + nε). 1≤k≤n
Repeating this argument with ε/2 replacing ε we get the following lemma:
w.p.1.
(27)
Proof. Writing ξk = log ak , applying Lemma 2.4, then exponentiating the resulting inequality, we get the claim. Using this lemma with λ = λ(A2 ) = λ2 in (26), substituting the resulting upper bound into (24), and using the arguments following (24), we get after some simplifications that for µ = max(λ1 , λ2 ) < 0, and for any ε there exists a random variable C 0 (ω), such that for all n ≥ 1 we have |xn2 | ≤ C 0 (ω)en(µ+ε) , and thus (22) follows. To estimate kA2n . . . A2k+1 k in the general case we need a simple observation that states that the subadditive process log kA2n . . . A2k+1 k can be majorated by a scalar-valued additive process modulo negligible error with growth rate arbitrarily close to λ2 = λ(A2 ) < 0.
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Lemma 2.6. Let A = (An ) be a strictly stationary, ergodic sequence of p × p random matrices such that E log+ kAn k < +∞. Then for any ε > 0 there exists a scalar-valued, stationary and ergodic process (ξn ), and a finite random variable c(ω), such that Eξn < λ(A) + ε, and for any 0 ≤ k < n log kAn . . . Ak+1 k ≤ c(ω) + ξn + . . . + ξk+1 + nε with probability 1. It follows that, with ak = eξk we have kAn . . . Ak+1 k ≤ C(ω)an . . . ak+1 enε , and here E log an < λ(A) + ε. The proof is essentially given in [11], and will be restyled and given in the Appendix. Applying the above result for A = A2 = (A2n ) in (24), the proof of Theorem 1.1 can be completed as in the scalar case.
Remark 3.2. Consider Eq. (1), satisfying the conditions of Section 1, except λ(A) < 0. A possible route to establish λ(A) < 0 would be to establish the existence of a causal stationary solution directly, and then use the following deep result of [6]. Theorem 3.1. Consider the linear stochastic system given by the state-space equation (1), where X n ∈ R p , (An , u n ) is a jointly strictly stationary sequence of random matrices of size p × ( p + 1), jointly satisfying condition (2). Let us assume that the sequence (An , u n ) is controllable in the sense that there is no proper subspace V ⊂ R p , such that A0 V + u 0 ⊂ V
w.p.1.
Then if (1) has a strictly stationary causal solution (X n ), then λ(A) < 0. Unfortunately, the direct verification of the existence of a causal stationary solution does not seem to be easy.
3. Discussion In this section we present a few remarks to highlight the delicacy of the details of the proof of the main result. Remark 3.1. To estimate kAn . . . Ak+1 k, see Lemma 2.6, an alternative, direct approach would be to use the F¨urstenberg–Kesten theorem starting at time k, and using the estimate that for all fixed ε > 0 we have kAn . . . Ak+1 k ≤ Ck+1 (ω)e(n−k)(λ+ε) , where λ = λ(A). Recall that Ck+1 (ω) can be defined as Ck+1 (ω) = eck+1 (ω) , where ck+1 (ω) = sup (log kAn . . . Ak+1 k − (n − k)(λ + ε)). n≥k+1
Using a representation of (An ) via a measure-preserving shift on Ω it is easily seen that (Ck+1 ) can be assumed to be a stationary sequence. To control the effect of (Ck+1 ) in (24) we would need to show that (Ck+1 ) is sub-exponential. One way to show this would be to show that E log+ Ck+1 (ω) < +∞.
(28)
Unfortunately, this inequality is not true in general. Indeed, consider a scalar-valued i.i.d. process An = an . Then λ = E log an , and for fixed ε > 0 we have log Ck+1 (ω) = sup
n X
(log a j − (λ + ε)).
n≥k+1 j=k+1
According to the Kiefer–Wolfowitz theorem (see [7], Chapter 10.4, Corollary 3) (28) holds if and only if E(log+ a j )2 < +∞. The same remark applies if we estimate kAn . . . Ak+1 k using the F¨urstenberg–Kesten theorem backwards in time. Then we get for fixed n, for all fixed ε > 0, and for all k ≤ n kAn . . . Ak+1 k ≤ Cn (ω)e(n−k)(λ+ε) , and here the growth rate of Cn (ω) is, in general, not under control.
Remark 3.3. We note in passing, that an equally non-trivial deterministic version of the above result, with An taking its values from a given set of matrices A, has been given in [12]. For more on this subject see [1,8]. Remark 3.4. The existence of the derivative process (X θ,n ) in an almost sure sense, and, as a byproduct, the existence of a strictly stationary causal solution of (9) has been proved by direct arguments in [3] in the case of GARCH processes using specific arguments. Applying Theorem 3.1, we could conclude, with some additional work, that the top-Lyapunov exponent of the matrix An 0 ¯ An = . Aθ,n An is negative. It is not clear if this line of proof can used in the general case. Remark 3.5. For the sake of completeness we present the main tool that is used in the proof of [15]: Lemma 3.1. Let A = (An ) be an i.i.d. sequence of p × p matrices with EkAn ks < ∞ for some s > 0. Then for the associated top-Lyapunov exponent we have λ(A) < 0 if and only if there exist c > 0, s > 0 and 0 < ρ < 1 such that EkAn . . . A1 ks ≤ cρ n
for n ≥ 1.
The proof is based on the observation that for any fixed l ≥ 1 the map u 7→ EkAl . . . A1 ku has first derivative with respect to u equal to E log kAl . . . A1 k at u = 0. Acknowledgement The authors acknowledge the support of the National Research Foundation of Hungary (OTKA) under Grant no. T 047193.
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Appendix Proof of Lemma 2.6. We follow the proof of [11]. For a fixed ε > 0 take an l such that 1 E log kAl . . . A1 k < λ + ε. l For any 0 ≤ l < n and 0 ≤ r < l take a cover of the index set {k + 1, . . . , n} by l-tuples of the form Iqr = {ql + r, . . . , ql + r + l − 1},
By assumption 1l Eξi < λ + ε, and (ξn ) is strictly stationary and ergodic, hence, applying Lemma 2.4 we get that with some c(ω) depending on ε we have log kAn . . . Ak+1 k ≤
n 1 X ξi + (c00 (ω) + εn) l i=k+1
≤ c(ω) + (n − k)(λ + ε) + nε + (c00 (ω) + εn) which implies the claim.
a.s.
References
and let q0 = q0 (r ) = min{q : ql + r > k + 1} q1 = q1 (r ) = max{q : ql + r ≤ n}. Set k¯ = q0l + r and n¯ = q1l + r − 1. Then kAn . . . Ak+1 k ≤ kAn . . . An+1 ¯ k ×kAn¯ . . . An−l+1 k . . . kAk+l−1 . . . Ak¯ kkAk−1 . . . Ak+1 k, ¯ ¯ ¯ thus log kAn . . . Ak+1 k can be estimated from above by log kAn . . . An+1 ¯ k+
qX 1 −1
log kAql+r +l−1 . . . Aql+r k
q=q0
+ log kAk−1 . . . Ak+1 k. ¯
(29)
By Lemma 2.1 we have that for any ε0 > 0 there exists a random variable c0 (ω) such that log kAk k ≤ c0 (ω) + ε0 k. Thus, since n − (n¯ + 1) ≤ l we have log kAn . . . An+1 ¯ k ≤ log kAn k + · · · 0 0 + log kAn+1 ¯ k ≤ l(c (ω) + ε n).
A similar inequality holds for log kAk−1 . . . Ak+1 k. Now define ¯ ξi = log kAi+l−1 . . . Ai k ηqr = log kAql+r +l−1 . . . Aql+r k. Note that ηqr = ξql+r . Then the middle term in (29) can be written as qX 1 −1 q=q0
ηqr =
qX 1 −1
ξql+r .
q=q0
Letting r run from 0 to (l − 1) and averaging (29) over r we get ! X 1 n−l+1 ξi + 2l(c0 (ω) + ε 0 n) log kAn . . . Ak+1 k ≤ l i=k+1 ! n 1 X ξi + 3l(c0 (ω) + ε 0 (n + l)) a.s., ≤ l i=k+1 since ξi ≤ l(c0 (ω) + ε0 (n + l)) for any i ≤ n. For fixed l taking ε 0 sufficiently small we can upper bound the last, residual term as c00 (ω) + εn for any prescribed ε.
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