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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 130915, 10 pages doi:10.1155/2010/130915 Research Article Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence Qing-pei Zang School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China Correspondence should be addressed to Qing-pei Zang, zqphunhu@yahoo.com.cn Received 4 May 2010; Revised 7 July 2010; Accepted 12 August 2010 Academic Editor: Soo Hak Sung Copyright q 2010 Qing-pei Zang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let {X n ; n ≥ 1} be a standardized non-stationary Gaussian sequence, and let denote S n   n k1 X k , σ n   VarS n . Under some additional condition, let the constants {u ni ;1≤ i ≤ n, n ≥ 1} satisfy  n i1 1 −Φu ni  → τ as n →∞for some τ ≥ 0andmin 1≤i≤n u ni ≥ clog n 1/2 ,forsomec>0, then, we have lim n →∞ 1/ log n  n k1 1/kI{∩ k i1 X i ≤ u ki ,S k /σ k ≤ x}  e −τ Φx almost surely for any x ∈ R,whereIA is the indicator function of the event A and Φx stands for the standard normal distribution function. 1. Introduction When {X, X n ; n ≥ 1} is a sequence of independent and identically distributed i.i.d. random variables and S n   n k1 X k ,n ≥ 1,M n  max 1≤k≤n X k for n ≥ 1. If EX0, VarX1, the so-called almost sure central limit theorem ASCLT has the simplest form as follows: lim n →∞ 1 log n n  k1 1 k I  S k √ k ≤ x  Φ  x  , 1.1 almost surely for all x ∈ R, where IA is the indicator function of the event A and Φx stands for the standard normal distribution function. This result was first proved independently by Brosamler 1 and Schatte 2 under a stronger moment condition; since then, this type of almost sure version was extended to different directions. For example, Fahrner and Stadtm ¨ uller 3 and Cheng et al. 4 extended this almost sure convergence for partial sums to the case of maxima of i.i.d. random variables. Under some natural conditions, they proved as follows: lim n →∞ 1 log n n  k1 1 k I  M k − b k a k ≤ x   G  x  a.s. 1.2 2 Journal of Inequalities and Applications for all x ∈ R, where a k > 0andb k ∈ R satisfy P  M k − b k a k ≤ x  −→ G  x  , as k −→ ∞ 1.3 for any continuity point x of G. In a related work, Cs ´ aki and Gonchigdanzan 5 investigated the validity of 1.2 for maxima of stationary Gaussian sequences under some mild condition whereas Chen and Lin 6 extended it to non-stationary Gaussian sequences. Recently, Dudzi ´ nski 7 obtained two-dimensional version for a standardized stationary Gaussian sequence. In this paper, inspired by the above results, we further study ASCLT in the joint version for a non-stationary Gaussian sequence. 2. Main Result Throughout this paper, let {X n ; n ≥ 1} be a non-stationary standardized normal sequence, and σ n   VarS n .Herea  b and a ∼ b stand for a  Ob and a/b → 1, respectively. Φx is the standard normal distribution function, and φx is its density function; C will denote a positive constant although its value may change from one appearance to the next. Now, we state our main result as follows. Theorem 2.1. Let {X n ; n ≥ 1} be a sequence of non-stationary standardized Gaussian variables with covariance matrix r ij  such that 0 ≤ r ij ≤ ρ |i−j| for i /  j,whereρ n ≤ 1 for all n ≥ 1 and sup s≥n  s−1 is−n ρ i  log n 1/2 /log log n 1ε ,ε>0. If the constants {u ni ;1 ≤ i ≤ n,n ≥ 1} satisfy  n i1 1 − Φu ni  → τ as n →∞for some τ ≥ 0 and min 1≤i≤n u ni ≥ clog n 1/2 ,forsomec>0, then lim n →∞ 1 log n n  k1 1 k I  k  i1  X i ≤ u ki  , S k σ k ≤ x   e −τ Φ  x  , 2.1 almost surely for any x ∈ R. Remark 2.2. The condition sup s≥n  s−1 is−n ρ i  log n 1/2 /log log n 1ε ,ε > 0 is inspired by a1 in Dudzi ´ nski 8, which is much more weaker. 3. Proof First, we introduce the following lemmas which will be used to prove our main result. Lemma 3.1. Under the assumptions of Theorem 2.1, one has  1≤i<j≤n r ij exp  − u 2 ni  u 2 nj 2  1  r ij   ≤ C  log log n  1ε . 3.1 Proof. This lemma comes from Chen and Lin 6. Journal of Inequalities and Applications 3 The following lemma is Theorem 2.1 and C orollary 2.1inLiandShao9. Lemma 3.2. (1) Let {ξ n } and {η n } be sequences of standard Gaussian variables with covariance matrices R 1 r 1 ij  and R 0 r 0 ij , respectively. Put ρ ij  max|r 1 ij |, |r 0 ij |. Then one has P ⎛ ⎝ n  j1  ξ j ≤ u j  ⎞ ⎠ − P ⎛ ⎝ n  j1  η j ≤ u j  ⎞ ⎠ ≤ 1 2π  1≤i<j≤n  arcsin  r 1 ij  − arcsin  r 0 ij   exp  − u 2 i  u 2 j 2  1  ρ ij   , 3.2 for any real numbers u i , i  1, 2, ,n. (2) Let {ξ n ; n ≥ 1} be standard Gaussian variables with r ij  Covξ i ,ξ j .Then       P ⎛ ⎝ n  j1  ξ j ≤ u j  ⎞ ⎠ − n  j1 P  ξ j ≤ u j        ≤ 1 4  1≤i<j≤n   r ij   exp  − u 2 i  u 2 j 2  1    r ij     , 3.3 for any real numbers u i , i  1, 2, ,n. Lemma 3.3. Let {X n } be a sequence of standard Gaussian variables and satisfy the conditions of Theorem 2.1, then for 1 ≤ k<n, one has P  n  ik1 { X i ≤ u ni } , S n σ n ≤ y  − P  n  i1 { X i ≤ u ni } , S n σ n ≤ y  ≤ k n  C  log log n  1ε 3.4 for any y ∈ R. Proof. By the conditions of Theorem 2.1, we have σ n   n  2  1≤i<j≤n r ij ≥ √ n, 3.5 then, for 1 ≤ i ≤ n,bysup s≥n  s−1 is−n ρ i  log n 1/2 /log log n 1ε ,ε>0, it follows that Cov  X i , S n σ n  ≤ 1 √ n  1 √ n n  k1 ρ k   log n  1/2 √ n  log log n  1ε . 3.6 Then, there exist numbers δ, n 0 , such that, for any n>n 0 , we have sup 1≤i≤n Cov  X i , S n σ n  <δ< 1 2 . 3.7 4 Journal of Inequalities and Applications We can write that L : P  n  ik1 { X i ≤ u ni } , S n σ n ≤ y  − P  n  i1 { X i ≤ u ni } , S n σ n ≤ y  ≤      P  n  ik1 { X i ≤ u ni } , S n σ n ≤ y  − P  n  ik1 { X i ≤ u ni }  P  Y n ≤ y             P  n  i1 { X i ≤ u ni } , S n σ n ≤ y  − P  n  i1 { X i ≤ u ni }  P  Y n ≤ y         P  n  ik1 { X i ≤ u ni }  − P  n  i1 { X i ≤ u ni }  : L 1  L 2  L 3 , 3.8 where {Y n } is a random variable, which has the same distribution as {S n /σ n },butit is independent of X 1 ,X 2 , ,X n . For L 1 ,L 2 , apply Lemma 3.2 1 with ξ i  X i ,i  1, ,n; ξ n1  S n /σ n , η j  X j ,j 1, ,n; η n1  Y n . Then r 1 ij  r 0 ij  r ij for 1 ≤ i<j≤ n and r 1 ij  CovX i ,S n /σ n ,r 0 ij  0for1≤ i ≤ n, j  n  1. Thus, we have for i  1, 2 L i  n  i1 Cov  X i , S n σ n  exp  − u 2 ni  y 2 2  1  Cov  X i ,S n /σ n   . 3.9 Since 3.5, 3.7 hold, we obtain L i   log n  1/2 √ n  log log n  1ε n  i1 exp  − u 2 ni 2  1  δ   . 3.10 Now define u n by 1 − Φu n 1/n. By the well-known fact 1 − Φ  x  ∼ φ  x  x ,x−→ ∞, 3.11 it is easy to see that exp  − u 2 n 2  ∼ √ 2πu n n ,u n ∼  2logn. 3.12 Journal of Inequalities and Applications 5 Thus, according to the assumption min 1≤i≤n u ni ≥ clog n 1/2 , we have u ni ≥ cu n for some c>0. Hence L i ≤  log n  1/2 √ n  log log n  1ε  1≤i≤n exp  − u 2 ni 2  1  δ   ≤ √ n  log n  1/2  log log n  1ε exp  − u 2 n 2  1  δ    √ n   2logn  2δ/1δ n 1/1δ  log log n  1ε    log n  2δ/1δ n 1/1δ−1/2  1 n δ  ,δ  > 0. 3.13 Now, we are in a position to estimate L 3 . Observe that L 3  P  n  ik1 { X i ≤ u ni }  − P  n  i1 { X i ≤ u ni }  ≤      P  n  ik1 { X i ≤ u ni }  − n  ik1 Φ  u ni             P  n  i1 { X i ≤ u ni }  − n  i1 Φ  u ni             n  ik1 Φ  u ni  − n  i1 Φ  u ni       : L 31  L 32  L 33 . 3.14 For L 33 , it follows that L 33  n  ik1 Φ  u ni   1 − k  i1 Φ  u ni    1 − Φ k  u n   1 −  1 − 1 n  k ≤ k n . 3.15 By Lemma 3.2 2, we have L 3i ≤ 1 4  1≤i<j≤n r ij exp  − u 2 ni  u 2 nj 2  1  r ij   ,i 1, 2. 3.16 Thus by Lemma 3.1 we obtain the desired result. 6 Journal of Inequalities and Applications Lemma 3.4. Let {X n } be a sequence of standard Gaussian variables satisfying the c onditions of Theorem 2.1, then for 1 ≤ k<n, any y ∈ R, one has      Cov  I  k  i1 { X i ≤ u ki } , S k σ k ≤ y  ,I  n  ik1 { X i ≤ u ni } , S n σ n ≤ y         k n  log n  1/2  log log n  1ε  1  log log n  1ε . 3.17 Proof. Apply Lemma 3.2 1 with ξ i  X i , 1 ≤ i ≤ k, ξ k1  S k /σ k ,ξ i1  X i ,k 1 ≤ i ≤ n, ξ n2  S n /σ n , η j  ξ j , 1 ≤ j ≤ k  1,η j  ξ j ,k 2 ≤ j ≤ n  2, where ξ k2 , ,ξ n2  has the same distribution as ξ k2 , ,ξ n2 , but it is independent of ξ k2 , ,ξ n2 . Then, r 1 ij  r 0 ij for 1 ≤ i<j≤ k  1ork  2 ≤ i<j≤ n  2; r 1 ij  r ij−1 ,r 0 ij  0for1≤ i ≤ k, k  2 ≤ j ≤ n  1; r 1 ij  Cov  X i , S n σ n  ,r 0 ij  0for1≤ i ≤ k, j  n  2; r 1 ij  Cov  X i , S k σ k  ,r 0 ij  0fork  1 ≤ i ≤ n, j  k  1; r 1 ij  Cov  S k σ k , S n σ n  ,r 0 ij  0fori  k  1,j n  2. 3.18 Thus, combined with 3.5 , 3.7, it follows that      Cov  I  k  i1 { X i ≤ u ki } , S k σ k ≤ y  ,I  n  ik1 { X i ≤ u ni } , S n σ n ≤ y             P  k  i1 { X i ≤ u ki } , n  ik1 { X i ≤ u ni } , S k σ k ≤ y, S n σ n ≤ y  −P  k  i1 { X i ≤ u ki } , S k σ k ≤ y  P  n  ik1 { X i ≤ u ni } , S n σ n ≤ y       ≤ 1 4  1≤i≤k  k1≤j≤n r ij exp  − u 2 ki  u 2 nj 2  1  r ij    1 4 k  i1 Cov  X i , S n σ n  exp  − u 2 ki  y 2 2  1  Cov  X i ,S n /σ n    1 4 n  ik1 Cov  X i , S k σ k  exp  − u 2 ni  y 2 2  1  Cov  X i ,S k /σ k    1 4 Cov  S k σ k , S n σ n  ≤ 1 4  1≤i≤k  k1≤j≤n r ij exp  − u 2 ki  u 2 nj 2  1  r ij    1 4 k  i1 Cov  X i , S n σ n  exp  − u 2 ki 2  1  δ    1 4 n  ik1 Cov  X i , S k σ k  exp  − u 2 ni 2  1  δ    1 4 Cov  S k σ k , S n σ n  : T 1  T 2  T 3  T 4 . 3.19 Journal of Inequalities and Applications 7 Using Lemma 3.1, we have T 1 ≤ C  log log n  1ε ,ε>0. 3.20 By the similar technique that was applied to prove 3.10,weobtain T 2  1 n α ,α>0. 3.21 For T 3 ,bysup s≥n  s−1 is−n ρ i  log n 1/2 /log log n 1ε ,ε>0, and 3.12, we have T 3  exp  − u 2 n 2  1  δ   n  ik1 Cov  X i , S k σ k   1 n 1/1δ n  ik1 Cov  X i , S k σ k   1 n 1/1δ 1 √ k n  ik1 Cov  X i ,S k   1 n 1/1δ 1 √ k k  j1 n  ik1 Cov  X i ,X j   1 n 1/1δ 1 √ k k  j1 n  i1 ρ i  √ k n 1/1δ  log n  1/2  log log n  1ε  1 n β ,β>0. 3.22 As to T 4 ,by3.5 and 3.6, we have T 4  1 σ k k  i1 Cov  X i , S n σ n    k n  log n  1/2  log log n  1ε . 3.23 Thus the proof of this lemma is completed. Proof of Theorem 2.1. First, by assumptions and Theorem 6.1.3 in Leadbetter et al. 10,we have P  n  i1  X i ≤ u ni   −→ e −τ . 3.24 8 Journal of Inequalities and Applications Let Y n denote a random variable which has the same distribution as S n /σ n , but it is independent of X 1 ,X 2 , ,X n , then by 3.10, we derive P  n  i1  X i ≤ u ni  , S n σ n ≤ y  − P  n  i1  X i ≤ u ni   P  Y n ≤ y  −→ 0, as n −→ ∞. 3.25 Thus, by the standard normal property of Y n , we have lim n →∞ P  n  i1  X i ≤ u ni  , S n σ n ≤ y   e −τ Φ  y  ,y∈ R. 3.26 Hence, to complete the proof, it is sufficient to show lim n →∞ 1 log n n  k1 1 k  I  k  i1  X i ≤ u ki  , S k σ k ≤ x  − P  k  i1  X i ≤ u ki  , S k σ k ≤ x   0a.s. 3.27 In order to show this, by Lemma 3.1 in Cs ´ aki and Gonchigdanzan 5, we only need to prove Var  1 log n n  k1 1 k I  k  i1  X i ≤ u ki  , S k σ k ≤ x   1  log log n  1ε , 3.28 for ε>0andanyx ∈ R.Letη k  I{  k i1 X i ≤ u ki ,S k /σ k ≤ x}−P{  k i1 X i ≤ u ki ,S k /σ k ≤ x}. Then Var  1 log n n  k1 1 k I  k  i1  X i ≤ u ki  , S k σ k ≤ x   E  1 log n n  k1 1 k η k  2  1 log 2 n n  k1 1 k 2 E   η k   2  2 log 2 n  1≤k<l≤n   E  η k η l    kl : S 1  S 2 . 3.29 Since |η k |≤2, it follows that S 1  1 log 2 n . 3.30 Journal of Inequalities and Applications 9 Now, we turn to estimate S 2 . Observe that f or l>k   E  η k η l          Cov  I  k  i1 { X i ≤ u ki } , S k σ k ≤ x  ,I  l  i1 { X i ≤ u li } , S l σ l ≤ x       ≤      Cov  I  k  i1 { X i ≤ u ki } , S k σ k ≤ x  ,I  l  i1 { X i ≤ u li } , S l σ l ≤ x  −I  l  ik1 { X i ≤ u li } , S l σ l ≤ x             Cov  I  k  i1 { X i ≤ u ki } , S k σ k ≤ x  ,I  l  ik1 { X i ≤ u li } , S l σ l ≤ x       ≤ E      I  l  i1 { X i ≤ u li } , S l σ l ≤ x  − I  l  ik1 { X i ≤ u li } , S l σ l ≤ x             Cov  I  k  i1 { X i ≤ u ki } , S k σ k ≤ x  ,I  l  ik1 { X i ≤ u li } , S l σ l ≤ x       : S 21  S 22 . 3.31 By Lemma 3.3, we have S 21 ≤ k l  C  log log l  1ε . 3.32 Using Lemma 3.4, it follows that S 22 ≤  k l  log l  1/2  log log l  1ε  C  log log l  1ε . 3.33 Hence for l>k, we have   E  η k η l    ≤ k l  C  log log l  1ε   k l  log l  1/2  log log l  1ε . 3.34 10 Journal of Inequalities and Applications Consequently S 2  1 log 2 n ⎛ ⎝  1≤k<l≤n 1 kl ⎛ ⎝ k l   k l  log l  1/2  log log l  1ε ⎞ ⎠ ⎞ ⎠   1≤k<l≤n 1 kl  log log l  1ε  1 log 2 n  1≤k<l≤n 1 l 2  1 log 2 n  log n  1/2  log log n  1ε n  l2 1 l 3/2 l−1  k1 1 √ k  1 log 2 n n  l3 1 l  log log l  1ε l−1  k1 1 k  1 log n  1  log n  log log n  1ε  1 log 2 n n  l3 log l l  log log l  1ε  1 log n  1  log log n  1ε . 3.35 Thus, we complete the proof of 3.28 by 3.30 and 3.35. 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