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Hindawi Publishing 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 Random 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: Ondˇ ej Doˇ ly r s´ 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 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 central limit theorem 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 for product of partial sums of i.i.d. random variables as follows. Theorem 1.1. Let {Xn , n ≥ 1} be a sequence of i.i.d. positive random variables with EX1 μ>0 σ 2 . Denote γ σ /μ the coeﬃcient of variation. Then for any real x and Var X1 ⎛ ⎞ √ 1/γ k k 1 n1⎝ i 1 Si ≤ x⎠ 1.1 a.s., lim I Fx k !μk n → ∞ ln n k k1 n k 1 Xk , I ∗ is the indicator function, F · is the distribution function of the random where Sn N variable e , and N is a standard normal variable. Recently, Jin 6 had proved that 1.1 holds under appropriate conditions for strongly mixing positive random variables and gave an ASCLT for product of partial sums of strongly mixing as follows.
2 Journal of Inequalities and Applications Theorem 1.2. Let {Xn , n ≥ 1} be a sequence of identically distributed positive strongly mixing n σ 2 , dk random variable with EX1 μ > 0 and Var X1 1/k, Dn k 1 dk . Denote by n γ σ /μ the coeﬃcient of variation, σn Var k 1 Sk − kμ /kσ and Bn Var Sn . Assume 2 2 2 Bn E|X1 |2 δ < ∞ for some δ > 0, 2 lim σ0 > 0, n→∞ n 1.2 σ2 2 −r inf n > 0. for some r > 1 αn On , δ n∈N n Then for any real x ⎛ ⎞ 1/γσk k 1n i 1 Si dk I ⎝ ≤ x⎠ a.s. lim Fx 1.3 n → ∞ Dn k !μk k1 The sequence {dk , k ≥ 1} in 1.3 is called weight. Under the conditions of Theorem 1.2, ∗ ∗ ∗ ∗ it is easy to see that 1.3 holds for every sequence dk with 0 ≤ dk ≤ dk and Dn k ≤ n dk → ∞ 7 . Clearly, the larger the weight sequence dk is, the stronger is the result 1.3 . α n In the following sections, let dk e ln k /k, 0 ≤ α < 1/2, Dn k 1 dk , “ ” denote the inequality “≤” up to some universal constant. We ﬁrst give an ASCLT for strongly mixing positive random variables. Theorem 1.3. Let {Xn , n ≥ 1} be a sequence of identically distributed positive strongly mixing σ 2 , dk and Dn as mentioned above. Denote random variable with EX1 μ > 0 and Var X1 n by γ σ /μ the coeﬃcient of variation, σn Var k 1 Sk − kμ /kσ and Bn Var Sn . Assume 2 2 that E|X1 |2 δ < ∞ for some δ > 0, 1.4 2 O n−r for some r > 1 αn , 1.5 δ 2 Bn 2 lim σ0 > 0, 1.6 n→∞ n 2 σn 1.7 inf > 0. n∈N n Then for any real x ⎛ ⎞ 1/γσk k 1n i 1 Si dk I ⎝ ≤ x⎠ a.s. lim Fx 1.8 n → ∞ Dn k !μk k1 In order to prove Theorem 1.3 we ﬁrst establish ASCLT for certain triangular arrays of n random variables. In the sequel we shall use the following notation. Let bk,n i k 1/i and k n i 1 bi,n for k ≤ n with bk,n Xk − μ /σ , k ≤ 1, Sn 2 2 sk,n 0 if k > n. Yk k 1 Yk and n Sn,n bk,n Yk . k1
Journal of Inequalities and Applications 3 In this setting we establish an ASCLT for the triangular array bk,n Yk . Theorem 1.4. Under the conditions of Theorem 1.3, for any real x 1n Sk,k ≤x Φx a.s., lim dk I 1.9 n → ∞ Dn σk k1 where Φ x 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 {Xn , n ≥ 1} be a sequence of strongly mixing random variables with zero mean, and let {ak,n , 1 ≤ k ≤ n, n ≥ 1} be a triangular array of real numbers. Assume that n a2 < ∞, max |ak,n | −→ 0 as n −→ ∞. sup 2.1 k,n 1≤k≤n n k1 If for a certain δ > 0, {|Xk |2 δ } is uniformly integrable, infk Var Xk > 0, ∞ n n2/δ α n < ∞, Var ak,n Xk 1, 2.2 n1 n1 then n d ak,n Xk − − N 0, 1 . −→ 2.3 k1 α n Lemma 2.2 see 9 . Let dk e ln k /k, 0 ≤ α < 1/2, Dn dk ; then k1 1−α α Dn ∼ C ln n 2.4 exp ln n , where C 1/α as 0 < α < 1/2, C 1 as α 0. Lemma 2.3 see 8 . Let {Xn , n ≥ 1} be a strongly mixing sequence of random variables such that supn E|Xn |2 δ < ∞ for a certain δ > 0 and every n ≥ 1. Then there is a numerical constant c δ depending only on δ such that for every n > 1 one has δ/ 2 δ nj n 2 ≤c δ 2/δ 2.5 sup Cov Xi , Xj i αi sup Xk 2 δ, j k ij1 i1 1/p E | Xk | p where Xk , p > 1. p
4 Journal of Inequalities and Applications Lemma 2.4 see 9 . Let {ξk , k ≥ 1} be a sequence of random variables, uniformly bounded below and with ﬁnite variances, and let {dk , k ≥ 1} be a sequence of positive number. Let for n ≥ 1, Dn n n k 1 dk and Tn k 1 dk ξk . Assume that 1/Dn Dn 1 Dn −→ ∞ −→ 1, 2.6 Dn as n → ∞. If for some ε > 0, C and all n ETn ≤ C ln−1−ε Dn , 2 2.7 then a.s. Tn −−→ 0 −− as n −→ ∞. 2.8 Lemma 2.5 see 10 . Let {Xn , n ≥ 1} be a strongly mixing sequence of random variables with zero mean and supn E|Xn |2 δ < ∞ for a certain δ > 0. Assume that 1.5 and 1.6 hold. Then |Sn | a.s. lim sup 1 2.9 n→∞ 2 2σ0 n ln ln n 2.2. Proof of Theorem 1.4 From the deﬁnition of strongly mixing we know that {Yk , k ≥ 1} remain to be a sequence of identically distributed strongly mixing random variable with zero mean and unit variance. Let ak,n bk,n /σn ; note that n k −1 n n k−1 1 2n − b1,n , n ≥ 1, 2 2.10 bk,n b1,n 2 b1,n 2 k k k 2i 1 k1 k 2 and via 1.7 we have 2 2n − b1,n bk,n n n < ∞, a2 sup sup sup k,n 2 n n k 1 σn n k1 n 2.11 bk,n ln n √ −→ 0, n −→ ∞. max |ak,n | max 1≤k≤n σn n 1≤k≤n From the deﬁnition of Yk and 1.4 we have that {|Yk |2 δ } is uniformly integrable; note that n n Var k 1 bk,n Yk 2 inf Var Yk E Y1 1 > 0, Var ak,n Yk 1, 2.12 2 σn k k1
Journal of Inequalities and Applications 5 and applying 1.5 ∞ ∞ n−r < ∞. n2/δ α n 2/δ 2.13 n1 n1 Consequently using Lemma 2.1, we can obtain Sn,n d − − N 0, 1 −→ as n −→ ∞, 2.14 σn which is equivalent to Sn,n −→ Ef N as n −→ ∞ 2.15 Ef σn for any bounded Lipschitz-continuous function f ; applying Toeplitz Lemma 1n Sk,k −→ Ef N as n −→ ∞. dk Ef 2.16 Dn k 1 σk We notice that 1.9 is equivalent to 1n Sk,k Φx lim dk f a.s. 2.17 n → ∞ Dn σk k1 for all bounded Lipschitz continuous f ; it therefore remains to prove that 1n Sk,k Sk,k a.s. − Ef −−→ 0, −− n −→ ∞. Tn dk f 2.18 Dn k 1 σk σk f Sk,k /σk − Ef Sk,k /σk , Let ξk 2 n ≤E 2 dk dl |E ξk ξl | E dk ξk dk dl ξk ξl 1≤k≤l≤n 1≤k≤l≤n k1 dk dl |E ξk ξl | dk dl |E ξk ξl | 2.19 1≤k≤l≤n 1≤k≤l≤n l≤2k l>2k T1,n T2,n . From Lemma 2.2, we obtain for some constant C1 1−1/α α e ln n ∼ C1 Dn ln Dn 2.20 .
6 Journal of Inequalities and Applications Using 2.20 and property of f , we have n 2k 1 1−1/α α α e ln n Dn e ln n 2 2.21 T1,n dk Dn ln Dn . l k1 l k We estimate now T2,n . For l > 2k, Sl,l − S2k,2k ··· bl,l Yl − b1,2k Y1 ··· b1,l Y1 b2,l Y2 b2,2k Y2 b2k,2k Y2k 2.22 ··· b2k 1,l S2k b2k 1,l Y2k 1 bl,l Yl . Notice that Sk,k Sl,l |Eξk ξl | Cov f ,f σk σl Sl,l − S2k,2k − b2k Sk,k Sl,l 1,l S2k ≤ Cov f −f ,f 2.23 σk σl σl Sl,l − S2k,2k − b2k Sk,k 1,l S2k Cov f ,f , σk σl and the properties of strongly mixing sequence imply Sl,l − S2k,2k − b2k Sk,k 1,l S2k Cov f ,f αk. 2.24 σk σl Applying Lemma 2.3 and 2.10 , 2k−1 2k 2k bi,2k EYi2 2 Var S2k,2k 2 bi,2k bj,2k Cov Yi , Yj i1 j 1i j 1 2k−1 2k 2k ≤ 2 2 bi,2k 2 bj,2k Cov Yi , Yj k, 2.25 i1 j1 ij1 2 2k−1 2k 2k 2k EYi2 Var S2k E Yi 2 Cov Yi , Yj k. i1 i1 i1ji1
Journal of Inequalities and Applications 7 Consequently, via the properties of f , the Jensen inequality, and 1.7 , Sl,l − S2k,2k − b2k Sk,k Sl,l 1,l S2k −f Cov f ,f σk σl σl 2 ES2k,2k E b2k 1,l S2k E S2k,2k b2k 1,l S2k 2 2.26 ≤ σl σl σl Var S2k β Var S2k,2k k b2k , 1,l σl σl l where 0 < β < 1/2. Hence for l > 2k we have β k |Eξk ξl | 2.27 αk . l Consequently, we conclude from the above inequalities that β k T2,n dk dl α k l 1≤k≤l≤n l>2k 2.28 β k dk dl α k dk dl T2,n,1 T2,n,2 . l 1≤k≤l≤n 1≤k≤l≤n l>2k l>2k Applying 1.5 and Lemma 2.2 we can obtain for any η > 0 n n n n −1−η −1−η T2,n,1 ≤ 2 dk dl α k ln Dn dk dl Dn ln Dn . 2.29 k 1l 1 k1 l1 Notice that β β k k T2,n,2 dk dl dk dl T2,n,2,1 T2,n,2,2 , l l 2.30 1≤k≤l≤n 1≤k≤l≤n l>2k l>2k 2/β 2/β l/k ≥ ln Dn l/k < ln Dn n n −2 −2 −2 T2,n,2,1 ≤ ≤ ln Dn 2 dk dl ln Dn dk dl Dn ln Dn . 2.31 1≤k≤l≤n k1 l1 l>2k
8 Journal of Inequalities and Applications 2/β max{l : k ≤ l ≤ n, l/k < ln Dn }, then Let n0 n0 n0 n n n 1 α α T2,n,2,2 ≤ dk dl ≤ e ln n dk ln n0 − ln 2k e ln n dk l 2.32 k 1 l 2k k1 l 2k k1 Dn ln1−1/α Dn ln ln Dn . α e ln n Dn ln ln Dn 2 By 2.21 , 2.29 , 2.31 , and 2.32 , for some ε > 0 such that 2 n 1 −1−ε 2 2.33 ETn E dk ξk ln Dn , 2 Dn k1 applying Lemma 2.4, we have a.s. Tn −−→ 0. −− 2.34 2.3. Proof of Theorem 1.3 Let Ck Sk /μk; we have 1n 1 n Sk 1n Sn,n Ck − 1 −1 bk,n Yk . 2.35 γσn k 1 γσn k 1 μk σn k 1 σn We see that 1.9 is equivalent to 1n k 1 Ci − 1 ≤ x Φx, a.s. ∀x. 2.36 lim dk I n → ∞ Dn γσk i1 k1 Note that in order to prove 1.8 it is suﬃcient to show that 1n k 1 ln Ci ≤ x Φx, a.s. ∀x. 2.37 lim dk I n → ∞ Dn γσk i1 k1 From Lemma 2.5, for suﬃciently large k, we have 1/2 ln ln k |Ck − 1| O . 2.38 k O x2 for |x| < 1/2, thus Since ln 1 x x n n n n ln ln k 2 ln Ck − Ck − 1 Ck − 1 ln n ln ln n a.s. 2.39 k k1 k1 k1 k 1
Journal of Inequalities and Applications 9 Hence for any ε > 0 and for suﬃciently large n, we have 1n 1n 1n Ck − 1 ≤ x − ε ≤I ln Ck ≤ x ≤I Ck − 1 ≤ x I ε 2.40 γσn k 1 γσn k 1 γσn k 1 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 central limit theorem,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 3, pp. 561–574, 1988. 2 P. Schatte, “On strong versions of the central limit theorem,” Mathematische Nachrichten, vol. 137, pp. 249–256, 1988. 3 M. T. Lacey and W. Philipp, “A note on the almost sure central limit theorem,” Statistics & Probability Letters, vol. 9, no. 3, pp. 201–205, 1990. 4 M. Peligrad and Q. M. Shao, “A note on the almost sure central limit theorem for 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 almost sure central limit theorem for the 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 variables,” Journal of Zhejiang University, vol. 34, no. 1, pp. 24–27, 2007. 7 I. Berkes and E. Cs´ ki, “A universal result in almost sure central limit theory,” Stochastic Processes and a Their Applications, vol. 94, no. 1, pp. 105–134, 2001. 8 M. Peligrad and S. Utev, “Central limit theorem for linear processes,” The Annals of Probability, vol. 25, no. 1, pp. 443–456, 1997. 9 F. Jonsson, Almost Sure Central Limit Theory, Uppsala University: Department of Mathematics, 2007. 10 L. Chuan-Rong and L. Zheng-Yan, Limit Theory for Mixing Dependent Random Variabiles, Science Press, Beijing, China, 1997.