Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 576301, 9 pages doi:10.1155/2011/576301 ResearchArticleAlmostSureCentralLimitTheoremforProductofPartialSumsofStronglyMixingRandom Variables Daxiang Ye and Qunying Wu College of Science, Guilin University of Technology, Guilin 541004, China Correspondence should be addressed to Daxiang Ye, 3040801111@163.com Received 19 September 2010; Revised 1 January 2011; Accepted 26 January 2011 Academic Editor: Ond ˇ rej Do ˇ sl ´ y Copyright q 2011 D. Ye and Q. Wu. 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. We give here an almostsurecentrallimittheoremforproductofsumsofstronglymixing positive random variables. 1. Introduction and Results In recent decades, there has been a lot of work on the almostsurecentrallimittheorem ASCLT, we can refer to Brosamler 1, Schatte 2, Lacey and Philipp 3, and Peligrad and Shao 4. Khurelbaatar and Rempala 5 gave an ASCLT forproductofpartialsumsof i.i.d. random variables as follows. Theorem 1.1. Let {X n ,n ≥ 1} be a sequence of i.i.d. positive random variables with EX 1 μ>0 and VarX 1 σ 2 . Denote γ σ/μ the coefficient of variation. Then for any real x lim n →∞ 1 ln n n k1 1 k I ⎛ ⎝ k i1 S i k!μ k 1/γ √ k ≤ x ⎞ ⎠ F x a.s., 1.1 where S n n k1 X k , I∗ is the indicator function, F· is the distribution function of the random variable e N , and N is a standard normal variable. Recently, Jin 6 had p roved that 1.1 holds under appropriate conditions forstronglymixing positive random variables and gave an ASCLT forproductofpartialsumsofstronglymixing as follows. 2 Journal of Inequalities and Applications Theorem 1.2. Let {X n ,n ≥ 1} be a sequence of identically distributed positive stronglymixingrandom variable with EX 1 μ>0 and VarX 1 σ 2 , d k 1/k, D n n k1 d k . Denote by γ σ/μ the coefficient of variation, σ 2 n Var n k1 S k − kμ/kσ and B 2 n VarS n . Assume E | X 1 | 2δ < ∞ for some δ>0, lim n →∞ B 2 n n σ 2 0 > 0, α n O n −r for some r>1 2 δ , inf n∈N σ 2 n n > 0. 1.2 Then for any real x lim n →∞ 1 D n n k1 d k I ⎛ ⎝ k i1 S i k!μ k 1/γσ k ≤ x ⎞ ⎠ F x a.s. 1.3 The sequence {d k ,k ≥ 1} in 1.3 is called weight. Under the conditions ofTheorem 1.2, it is easy to see that 1.3 holds for every sequence d ∗ k with 0 ≤ d ∗ k ≤ d k and D ∗ n k≤n d ∗ k →∞ 7. Clearly, the larger the weight sequence d k is, the stronger is the result 1.3. In the following sections, let d k e ln k α /k,0≤ α<1/2,D n n k1 d k ,“” denote the inequality “≤” up to some universal constant. We first give an ASCLT forstronglymixing positive random variables. Theorem 1.3. Let {X n ,n ≥ 1} be a sequence of identically distributed positive stronglymixingrandom variable with EX 1 μ>0 and VarX 1 σ 2 , d k and D n as mentioned above. Denote by γ σ/μ the coefficient of variation, σ 2 n Var n k1 S k − kμ/kσ and B 2 n VarS n . Assume that E | X 1 | 2δ < ∞ for some δ>0, 1.4 α n O n −r for some r>1 2 δ , 1.5 lim n →∞ B 2 n n σ 2 0 > 0, 1.6 inf n∈N σ 2 n n > 0. 1.7 Then for any real x lim n →∞ 1 D n n k1 d k I ⎛ ⎝ k i1 S i k!μ k 1/γσ k ≤ x ⎞ ⎠ F x a.s. 1.8 In order to prove Theorem 1.3 we first establish ASCLT for certain triangular arrays ofrandom variables. In the sequel we shall use the following notation. Let b k,n n ik 1/i and s 2 k,n k i1 b 2 i,n for k ≤ n with b k,n 0ifk>n. Y k X k − μ/σ, k ≤ 1, S n n k1 Y k and S n,n n k1 b k,n Y k . Journal of Inequalities and Applications 3 In this setting we establish an ASCLT for the triangular array b k,n Y k . Theorem 1.4. Under the conditions ofTheorem 1.3, for any real x lim n →∞ 1 D n n k1 d k I S k,k σ k ≤ x Φ x a.s., 1.9 where Φx is the standard normal distribution function. 2. The Proofs 2.1. Lemmas To prove theorems, we need the following lemmas. Lemma 2.1 see 8. Let {X n ,n ≥ 1} be a sequence ofstronglymixingrandom variables with zero mean, and let {a k,n , 1 ≤ k ≤ n, n ≥ 1} be a triangular array of real numbers. Assume that sup n n k1 a 2 k,n < ∞, max 1≤k≤n | a k,n | −→ 0 as n −→ ∞. 2.1 If for a certain δ>0, {|X k | 2δ } is uniformly integrable, inf k VarX k > 0, ∞ n1 n 2/δ α n < ∞, Var n n1 a k,n X k 1, 2.2 then n k1 a k,n X k d −−−→ N 0, 1 . 2.3 Lemma 2.2 see 9. Let d k e ln k α /k, 0 ≤ α<1/2,D n n k1 d k ;then D n ∼ C ln n 1−α exp ln n α , 2.4 where C 1/α as 0 <α<1/2,C 1 as α 0. Lemma 2.3 see 8. Let {X n ,n ≥ 1} be a stronglymixing sequence ofrandom variables such that sup n E|X n | 2δ < ∞ for a certain δ>0 and every n ≥ 1. Then there is a numerical constant cδ depending only on δ such that for every n>1 one has sup j nj ij1 Cov X i ,X j ≤ c δ n i1 i 2/δ α i δ/2δ sup k X k 2 2δ , 2.5 where X k p E|X k | p 1/p ,p>1. 4 Journal of Inequalities and Applications Lemma 2.4 see 9. Let {ξ k ,k ≥ 1} be a sequence ofrandom variables, uniformly bounded below and with finite variances, and let {d k ,k ≥ 1} be a sequence of positive number. Let for n ≥ 1,D n n k1 d k and T n 1/D n n k1 d k ξ k . Assume that D n −→ ∞ D n1 D n −→ 1, 2.6 as n →∞.Ifforsomeε>0, C and all n ET 2 n ≤ C ln −1−ε D n , 2.7 then T n a.s. −−−−→ 0 as n −→ ∞. 2.8 Lemma 2.5 see 10. Let {X n ,n≥ 1} be a stronglymixing sequence ofrandom variables with zero mean and sup n E|X n | 2δ < ∞ for a certain δ>0. Assume that 1.5 and 1.6 hold. Then lim sup n →∞ | S n | 2σ 2 0 n ln ln n 1 a.s. 2.9 2.2. Proof ofTheorem 1.4 From the definition ofstronglymixing we know that {Y k ,k ≥ 1} remain to be a sequence of identically distributed stronglymixingrandom variable with zero mean and unit variance. Let a k,n b k,n /σ n ;notethat n k1 b 2 k,n b 1,n 2 n k2 k−1 i1 1 k b 1,n 2 n k2 k − 1 k 2n − b 1,n ,n≥ 1, 2.10 and via 1.7 we have sup n n k1 a 2 k,n sup n n k1 b 2 k,n σ 2 n sup n 2n − b 1,n n < ∞, max 1≤k≤n | a k,n | max 1≤k≤n b k,n σ n ln n √ n −→ 0,n−→ ∞. 2.11 From the definition of Y k and 1.4 we have that {|Y k | 2δ } is uniformly integrable; note that inf k Var Y k EY 2 1 1 > 0, Var n k1 a k,n Y k Var n k1 b k,n Y k σ 2 n 1, 2.12 Journal of Inequalities and Applications 5 and applying 1.5 ∞ n1 n 2/δ α n ∞ n1 n −r2/δ < ∞. 2.13 Consequently using Lemma 2.1, we can obtain S n,n σ n d −−−→N 0, 1 as n −→ ∞, 2.14 which is equivalent to Ef S n,n σ n −→ Ef N as n −→ ∞ 2.15 for any bounded Lipschitz-continuous function f; applying Toeplitz Lemma 1 D n n k1 d k Ef S k,k σ k −→ Ef N as n −→ ∞. 2.16 We notice that 1.9 is equivalent to lim n →∞ 1 D n n k1 d k f S k,k σ k Φ x a.s. 2.17 for all bounded Lipschitz continuous f; it therefore remains to prove that T n 1 D n n k1 d k f S k,k σ k − Ef S k,k σ k a.s. −−−−→ 0,n−→ ∞. 2.18 Let ξ k fS k,k /σ k − EfS k,k /σ k , E n k1 d k ξ k 2 ≤ E 2 1≤k≤l≤n d k d l ξ k ξ l 1≤k≤l≤n d k d l | E ξ k ξ l | 1≤k≤l≤n l≤2k d k d l | E ξ k ξ l | 1≤k≤l≤n l>2k d k d l | E ξ k ξ l | T 1,n T 2,n . 2.19 From Lemma 2.2, we obtain for some constant C 1 e ln n α ∼ C 1 D n ln D n 1−1/α . 2.20 6 Journal of Inequalities and Applications Using 2.20 and property of f, we have T 1,n e ln n α n k1 d k 2k lk 1 l D n e ln n α D 2 n ln D n 1−1/α . 2.21 We estimate now T 2,n . For l>2k, S l,l − S 2k,2k b 1,l Y 1 b 2,l Y 2 ··· b l,l Y l − b 1,2k Y 1 b 2,2k Y 2 ··· b 2k,2k Y 2k b 2k1,l S 2k b 2k1,l Y 2k1 ··· b l,l Y l . 2.22 Notice that | Eξ k ξ l | Cov f S k,k σ k ,f S l,l σ l ≤ Cov f S k,k σ k ,f S l,l σ l − f S l,l − S 2k,2k − b 2k1,l S 2k σ l Cov f S k,k σ k ,f S l,l − S 2k,2k − b 2k1,l S 2k σ l , 2.23 and the properties ofstronglymixing sequence imply Cov f S k,k σ k ,f S l,l − S 2k,2k − b 2k1,l S 2k σ l α k . 2.24 Applying Lemma 2.3 and 2.10, Var S 2k,2k 2k i1 b 2 i,2k EY 2 i 2 2k−1 j1 2k ij1 b i,2k b j,2k Cov Y i ,Y j ≤ 2k i1 b 2 i,2k 2 2k−1 j1 b 2 j,2k 2k ij1 Cov Y i ,Y j k, Var S 2k E 2k i1 Y i 2 2k i1 EY 2 i 2 2k−1 i1 2k ji1 Cov Y i ,Y j k. 2.25 Journal of Inequalities and Applications 7 Consequently, via the properties of f,the Jensen inequality, and 1.7, Cov f S k,k σ k ,f S l,l σ l − f S l,l − S 2k,2k − b 2k1,l S 2k σ l E S 2k,2k b 2k1,l S 2k σ l ≤ ES 2 2k,2k σ l E b 2k1,l S 2k 2 σ l Var S 2k,2k σ l b 2k1,l Var S 2k σ l k l β , 2.26 where 0 <β<1/2. Hence for l>2k we have | Eξ k ξ l | α k k l β . 2.27 Consequently, we conclude from the above inequalities that T 2,n 1≤k≤l≤n l>2k d k d l α k k l β 1≤k≤l≤n l>2k d k d l α k 1≤k≤l≤n l>2k d k d l k l β T 2,n,1 T 2,n,2 . 2.28 Applying 1.5 and Lemma 2.2 we can obtain for any η>0 T 2,n,1 ≤ n k1 n l1 d k d l α k ln D n −1−η n k1 d k n l1 d l D 2 n ln D n −1−η . 2.29 Notice that T 2,n,2 1≤k≤l≤n l>2k l/k≥ ln D n 2/β d k d l k l β 1≤k≤l≤n l>2k l/k< ln D n 2/β d k d l k l β T 2,n,2,1 T 2,n,2,2 , 2.30 T 2,n,2,1 ≤ 1≤k≤l≤n l>2k d k d l ln D n −2 ≤ ln D n −2 n k1 d k n l1 d l D 2 n ln D n −2 . 2.31 8 Journal of Inequalities and Applications Let n 0 max{l : k ≤ l ≤ n, l/k < ln D n 2/β }, then T 2,n,2,2 ≤ n k1 n 0 l2k d k d l ≤ e ln n α n k1 d k n 0 l2k 1 l e ln n α n k1 d k ln n 0 − ln 2k e ln n α D n ln ln D n D 2 n ln 1−1/α D n ln ln D n . 2.32 By 2.21, 2.29, 2.31,and2.32, for some ε>0 such that ET 2 n 1 D 2 n E n k1 d k ξ k 2 ln D n −1−ε , 2.33 applying Lemma 2.4, we have T n a.s. −−−−→ 0. 2.34 2.3. Proof ofTheorem 1.3 Let C k S k /μk; we have 1 γσ n n k1 C k − 1 1 γσ n n k1 S k μk − 1 1 σ n n k1 b k,n Y k S n,n σ n . 2.35 We see that 1.9 is equivalent to lim n →∞ 1 D n n k1 d k I 1 γσ k k i1 C i − 1 ≤ x Φ x , a.s. ∀x. 2.36 Note that in order to prove 1.8 it is sufficient to show that lim n →∞ 1 D n n k1 d k I 1 γσ k k i1 ln C i ≤ x Φ x , a.s. ∀x. 2.37 From Lemma 2.5,forsufficiently large k, we have | C k − 1 | O ln ln k k 1/2 . 2.38 Since ln1 xx Ox 2 for |x| < 1/2, thus n k1 ln C k − n k1 C k − 1 n k1 C k − 1 2 n k1 ln ln k k ln n ln ln n a.s. 2.39 Journal of Inequalities and Applications 9 Hence for any ε>0andforsufficiently large n, we have I 1 γσ n n k1 C k − 1 ≤ x − ε ≤ I 1 γσ n n k1 ln C k ≤ x ≤ I 1 γσ n n k1 C k − 1 ≤ x ε 2.40 and thus 2.36 implies 2.37. Acknowledgment This work is supported by the National Natural Science Foundation of China 11061012, Innovation Project of Guangxi Graduate Education 200910596020M29. References 1 G. A. Brosamler, “An almost everywhere centrallimit theorem,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 3, pp. 561–574, 1988. 2 P. Schatte, “On strong versions of the centrallimit theorem,” Mathematische Nachrichten, vol. 137, pp. 249–256, 1988. 3 M. T. Lacey and W. Philipp, “A note on the almostsurecentrallimit theorem,” Statistics & Probability Letters, vol. 9, no. 3, pp. 201–205, 1990. 4 M. Peligrad and Q. M. Shao, “A note on the almostsurecentrallimittheoremfor weakly dependent random variables,” Statistics & Probability Letters, vol. 22, no. 2, pp. 131–136, 1995. 5 G. Khurelbaatar and G. Rempala, “A note on the almostsurecentrallimittheoremfor the productofpartial sums,” Applied Mathematics Letters, vol. 19, pp. 191–196, 2004. 6 J. S. Jin, “An almostsurecentrallimittheoremfor the productofpartialsumsofstrongly missing random variables,” Journal of Zhejiang University, vol. 34, no. 1, pp. 24–27, 2007. 7 I. Berkes and E. Cs ´ aki, “A universal result in almostsurecentrallimit theory,” Stochastic Processes and Their Applications, vol. 94, no. 1, pp. 105–134, 2001. 8 M. Peligrad and S. Utev, “Central limittheoremfor linear processes,” The Annals of Probability, vol. 25, no. 1, pp. 443–456, 1997. 9 F. Jonsson, AlmostSureCentralLimit Theory, Uppsala University: Department of Mathematics, 2007. 10 L. Chuan-Rong and L. Zheng-Yan, Limit Theory forMixing Dependent Random Variabiles, Science Press, Beijing, China, 1997. . Corporation Journal of Inequalities and Applications Volume 2011, Article ID 576301, 9 pages doi:10.1155/2011/576301 Research Article Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing. almost sure central limit theorem for product of sums of strongly mixing positive random variables. 1. Introduction and Results In recent decades, there has been a lot of work on the almost sure. product of partial sums, ” Applied Mathematics Letters, vol. 19, pp. 191–196, 2004. 6 J. S. Jin, “An almost sure central limit theorem for the product of partial sums of strongly missing random