Commun. Korean Math. Soc. 21 (2006), No. 4, pp. 779 786 FUNCTIONAL CENTRAL LIMIT THEOREMS FOR MULTIVARIATE LINEAR PROCESSES GENERATED BY DEPENDENT RANDOM VECTORS Mi-Hwa Ko Abstract. Let X t be an m-dimensional linear process defined by X t = A j Z t j, t = 1, 2,..., where {Z t } is a sequence of m-dimensional random vectors with mean 0 : m 1 and positive definite covariance matrix Γ : m m and {A j } is a sequence of coefficient matrices. In this paper we give sufficient conditions so that [ns] t=1 Xt (properly normalized) converges weakly to Wiener measure if the corresponding result for [ns] t=1 Z t is true. 1. Introduction Consider m-dimensional linear process of the form (1.1) X t = A j Z t j, where the innovation {Z t } is a sequence of m-dimensional random vectors with mean 0 : m 1 and positive definite covariance matrix Γ : m m. Throughout we shall assume that (1.2) A j < and A j O m m, where for any m m, m 1, matrix A = (a ij ), i, j = 1,..., m, A = m m a ij and O m m denotes the m m zero matrix. Let W m denote Wiener measure on D m [0, 1], the space of all real valued functions Received July 27, 2006. Revised September 18, 2006. 2000 Mathematics Subject Classification: 60F05, 60G10. Key words and phrases: functional central limit theorem, Linear process, moving average process, negatively associated, martingale difference. This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-037-C00007).
780 Mi-Hwa Ko on [0,1] that are right continuous and have finite left limits, endowed with the sup norm(see, e.g., [3], [10]). Further, let ( ( ) (1.3) T = A j )Γ A j, S n = n t=1 X t, n 1(S 0 = O) and define for n 1 the stochastic process ξ n by (1.4) ξ n (s) = n 1 2 S[ns] 0 s 1. In this paper we give sufficient conditions so that [ns] t=1 X t (properly normalized) converges weakly to Wiener measure if the corresponding result for [ns] t=1 Z t is true. As applications we also discuss functional central limit theorems for linear processes generated by martingale difference and negatively associated random vectors. 2. Main results Theorem 2.1. Let X t satisfy model (1.1) and be a positive constant sequence satisfying that as n. Assume that {A j } satisfies (1.2) and {Z t } is any random vector sequence satisfying (2.1) sup E max m Z k+j 2 Cd 2 (n) for every n 1 j 1 m n and, as n, (2.2) Then, (2.3) k=1 1 max Z k p 0. n k n k n (s) 1 Z t W m (s) implies t=1 k n (s) 1 t=1 X t BW m (s), where k n (s) = sup{m : d 2 (m) sd 2 (n)} and B = k=0 A k. Theorem 2.1 can be applied to many important cases, such as whether innovation {Z t } is martingale difference or negatively associated sequence. In the following, we will derive corollaries of Theorem 2.1. We note that Corollary 2.3 below is Theorem 1(i) of [6] and Corollary 2.8 is a new result.
Functional central limit theorems 781 Definition 2.2. Let {Z k } be a random vector sequence. We say that {Z k } is a martingale difference sequence if E(Z k F k 1 ) = 0, a.s. k = 0, ±1, ±2,..., where F k = σ{z i, i k}. Corollary 2.3. Define X t as in (1.1) and ξ n as in (1.4), respectively. Let {Z t } be a sequence of m-dimensional martingale difference vectors with E(Z t F t 1 ) = 0 a.s. and Γ t denote the conditional covariance matrix of Z t, E(Z t Z t F t 1 ) = Γ t a.s., such that 1 n n t=1 Γ t p Γ, where F t is sub-σ-algebra generated by Z u, u t and the prime denotes transpose and Γ is a positive definite(d.f.) non random matrix. Assume that sup t E Z t 2 < and 1 n n t=1 E(Z tz ti(z t Z t > nɛ) F t 1 ) p 0 as n for every ɛ > 0, where I( ) denotes the indicator function. Then, ξ n W m, where W m is a Wiener measure with covariance matrix T = ( A j)γ( A j). Proof. Define for n 1 the stochastic process η n by [ns] (2.4) η n (s) = n 1 2 Z i, 0 s 1. It follows from the multivariate version of Theorem 1 of [2] or Theorem 2 of [1] that η n (s) converges weakly to Wiener measure with covariance matrix Γ (c.f. Theorem 3.1 of [8]). On the other hand, it follows from Doob s maximal inequality and sup t E Z t 2 < that for every n 1 (2.5) sup E max j and (2.6) 1 m n ( m k=1 Z k+j ) 2 C 1 n sup j 1 max Z k p 0. n n k n sup E Z k+j 2 C 2 n k Hence, corollary 2.3 follows immediately from Theorem 2.1 with = n. Definition 2.4. Let {Z i, 1 i n} be a sequence of m-dimensional random vectors. They are said to be negatively associated(na) for every pair of disjoint subsets A and B of {1,..., n} Cov(f(Z i, i A), g(z j, j B)) 0 whenever f and g are coordinatewise increasing and the covariance exists. An infinite family is negatively associated if every finite subfamily is negatively associated. Lemma 2.5. Let r 2 and let {Z i, 1 i n} be a sequence of m-dimensional negatively associated random vectors with EZ i = 0 and
782 Mi-Hwa Ko E Z i r <, where Z i = (Zi1 2 + + Z2 im ) 1 2. Then there exists a constant 0 < A r < such that k n n (2.7) E max Z i r A r m r {( E Z i 2 ) r 2 + E Z i r }. Proof. Note that k (2.8) max Z i m k max Z ij and by the result in [11] we have k n n E max Z ij r A r {( E(Z ij ) 2 ) r 2 + E Z ij r } (2.9) n n A r {( E Z i 2 ) r 2 + E Z i r }. Hence, from (2.8) and (2.9) equation (2.7) follows. Lemma 2.6. Let {Z i, 1 i n} be a sequence of m-dimensional negatively associated random vectors with E(Z i ) = 0 and E Z i 2 <. Then for all x > 0 and a > 0, (2.10) k P ( max Z i mx) 2mP ( max Z k > a) + 4m exp( 8 n E Z i 2 ) n + 4m( E Z i 2 4(xa + n E Z i 2 ) )x/(12a). Proof. From (2.8) and the result of [11], (2.10) follows easily. Theorem 2.7. Let {Z i, i 1} be a strictly stationary sequence of m-dimensional negatively associated random vectors with E(Z 1 ) = 0 and E Z 1 2 <. Define, for t [0, 1], n 1 ξ n (t) = n 1 [nt] 2 Z i. If E Z 1 2 +2 m i=2 E(Z 1jZ ij ) = σ 2 <, then, as n, ξ n B m, where B m is an m-dimensional Wiener measure with covariance matrix Γ = (σ kj ) and σ kj = E(Z 1k Z 1j ) + i=2 [E(Z 1kZ ij ) + E(Z 1j Z ik )]. Proof. By means of the simple device due to Cramer Wold (see [3], [4]), from the Newman s central limit theorem for negatively associated x 2
Functional central limit theorems 783 random variables(see [9]) we obtain n 1 2 n Z i D N(0, Γ), where N(0, Γ) denotes an m-dimensional normal random vector and the symbol D indicates convergence in distribution. Hence, as in the proof of Theorem 2 of [5] on weakly associated random vectors, the limit point of ξ n ( ) is an m-dimensional Wiener measure with covariance matrix Γ = (σ kj ). It remains to verify the tightness of ξ n ( ) (see Theorem 15.1 of [3]). By Theorem 8.4 of [3] we only need to show that for any ɛ > 0, there exist a positive number λ and an integer n such that for every n n 0 k (2.11) P ( max Z i > λn 1 2 ) m 3 ɛλ 2. Applying Lemma 2.6 with λ = mλ, x = λ n 1 2 and a = λ n 1 2 /48 k P ( max Z i > λn 1 2 ) k = P ( max Z i > mλ n 1 2 ) 2mP ( max Z k > λ n 1 2 /48) +4m exp( λ 2 n 8nE Z 1 2 ) + 4m( ne Z 1 2 4(nE Z 1 2 + λ 2 n/48) )4 2m(48) 2 λ 2 E Z 1 2 I{ Z 1 > λ n 1 2 /48} +4m exp( λ 2 8E Z 1 2 ) + 4m(12E Z 1 2 mɛλ 2 = m 3 ɛλ 2 provided that λ is sufficiently large. This proves (2.11), and hence the proof of Theorem 2.7 is complete. Corollary 2.8. Let {Z i, i 1} be a strictly stationary negatively associated sequence of m-dimensional random vectors centered at expectations and E Z 1 2 < and X t be defined as in (1.1). Let the stochastic process ξ n be defined as in (1.4). Assume (1.2) and E Z 1 2 + 2 m i=2 E(Z 1jZ ij ) = σ 2 < hold. Then ξ n W m. Proof. First note that ξ n (s) = n 1 [ns] 2 Z i converses weakly to Wiener measure B m with covariance matrix Γ by Theorem 2.7. On the λ 2 ) 4
784 Mi-Hwa Ko other hand, it follows from Lemma 2.5 and the condition E Z 1 2 < that (2.5) and (2.6) hold. Hence, Corollary 2.8 follows immediately from Theorem 2.1 with = n. 3. Proof of Theorem 2.1 For every fixed l 1, put (3.1) X (l) 1j = (3.2) l k=0 A k Z j k and X (l) 2j = k=l+1 A k Z j k. From the idea in [7] (p.320) we obtain that for any m 1, = = m l X (l) 1j = m m A k k=0 l m A k k=0 k=0 Z j + l A k Z j k l Z 1 s s=1 j=s Z j + R(m, l), (say). l l 1 A j + Therefore, it follows that for every fixed l 1, l Z m s A j s=0 j=s+1 (3.3) k n (s) 1 X t = ( t=1 l 1 A k ) k=0 k n (s) + 1 k n(s) X (l) 2j. Z j + 1 R(k n(s), l) By (3.3), Theorem 4.1 given in [3] (p.25) and noting that l k=0 A k B as l, to prove (2.3), it suffices to show that for any δ > 0, (3.4) lim sup P { sup R(k n (t), l) δ} = 0, n 0 t 1 for every fixed l 1 and (3.5) lim lim sup l n k n (t) P { sup 0 t 1 X (l) 2j δ} = 0.
Functional central limit theorems 785 By condition (2.2) since k=0 A k <, as n, 1 sup R(k n (s), l) 0 t 1 l 1 max Z j l j n s=0 ( l A j + j=s j=s+1 A u ) P 0 and hence (3.4) holds. Noting that m X(l) 2j = k=l+1 A m k Z j k for any m 1, by applying Hölder inequality and (2.1), we have k n (t) E sup X (l) 2j 2 ( 1 t 1 k=l+1 Cd 2 (n)( A k ) 2 E max m Z j k 2 1 m n k=l+1 A k ) 2. Hence, (3.5) follows immediately from the Markov inequality and k=l+1 A k 0 as l. The proof of Theorem 2.1 is complete. References [1] D. J. Aldous and G. K. Eagleson, On mixing and stability of limit theorems, Ann. Probability. 6 (1978), 325 331. [2] G. J. Babu and M. Ghosh, A ranldom functional central limit theorem for martingales, Acta. Math. Scie. Hungar. 27 (1976), 301 306. [3] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. [4] P. J. Brockwell and R. A. Davis, Time Series : Theory and Methods 2nd ed., Springer, New York, 1987. [5] R. M. Burton, A. R. Darbrowski, and H. Dehling, An invariance principle for weakly associated random vectors, Stoc. Proc. and Appl. 23 (1986), 301 306. [6] I. Fakhre-Zakeri and S. Lee, On functional central limit theorems for multivariate linear processes with applications to sequential estimation, J. Stat. Plan. Inf. 3 (2000), 11 23. [7] W. A. Fuller, Introduction to Statistical Time-Series, 2nd Edition. Wiley Inc, New York, 1996. [8] T. S. Kim, Y. K. Choi, and M. H. Ko, The random functional central limit theorem for multivariate martingale difference, Taiwanese J. Math. accepted, (2006). [9] C. M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables, in: Tong, Y.L., ed., IMS Lecture Notes- Monograph series. 5 (1984), 127 140. [10] D. Pollard, Convergence of Stochastic Processes, Wiley, New York, 1984.
786 Mi-Hwa Ko [11] Q. M. Shao, A comparison theorem on maximum inequalities between negatively associated and independent random variables, J. Theor. Probab. 13 (2000), 343 356. Department of Mathematics WonKwang University Jeonbuk 570-749, Korea E-mail: songhack@wonkwang.ac.kr