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: 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 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 {X n ,n ≥ 1} be a sequence of i.i.d. positive random variables with EX 1  μ>0 and VarX 1 σ 2 . Denote γ  σ/μ the coefficient of variation. Then for any real x lim n →∞ 1 ln n n  k1 1 k I ⎛ ⎝   k i1 S i k!μ k  1/γ √ k ≤ x ⎞ ⎠  F  x  a.s., 1.1 where S n   n k1 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 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 {X n ,n ≥ 1} be a sequence of identically distributed positive strongly mixing random variable with EX 1  μ>0 and VarX 1 σ 2 , d k  1/k, D n   n k1 d k . Denote by γ  σ/μ the coefficient of variation, σ 2 n  Var  n k1 S k − kμ/kσ and B 2 n  VarS n . Assume E | X 1 | 2δ < ∞ 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  k1 d k I ⎛ ⎝   k i1 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 of Theorem 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 k1 d k ,“” denote the inequality “≤” up to some universal constant. We first give an ASCLT for strongly mixing positive random variables. Theorem 1.3. Let {X n ,n ≥ 1} be a sequence of identically distributed positive strongly mixing random variable with EX 1  μ>0 and VarX 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 k1 S k − kμ/kσ and B 2 n  VarS n . Assume that E | X 1 | 2δ < ∞ 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  k1 d k I ⎛ ⎝   k i1 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 of random variables. In the sequel we shall use the following notation. Let b k,n   n ik 1/i and s 2 k,n   k i1 b 2 i,n for k ≤ n with b k,n  0ifk>n. Y k X k − μ/σ, k ≤ 1,  S n   n k1 Y k and S n,n   n k1 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 of Theorem 1.3, for any real x lim n →∞ 1 D n n  k1 d k I  S k,k σ k ≤ x  Φ  x  a.s., 1.9 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 {X n ,n ≥ 1} be a sequence of strongly mixing random 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  k1 a 2 k,n < ∞, max 1≤k≤n | a k,n | −→ 0 as n −→ ∞. 2.1 If for a certain δ>0, {|X k | 2δ } is uniformly integrable, inf k VarX k  > 0, ∞  n1 n 2/δ α  n  < ∞, Var  n  n1 a k,n X k   1, 2.2 then n  k1 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 k1 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 strongly mixing sequence of random variables such that sup n E|X n | 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 sup j nj  ij1   Cov  X i ,X j    ≤ c  δ   n  i1 i 2/δ α  i   δ/2δ sup k  X k  2 2δ , 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 of random 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 k1 d k and T n 1/D n   n k1 d k ξ k . Assume that D n −→ ∞ D n1 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 strongly mixing sequence of random variables with zero mean and sup n E|X n | 2δ < ∞ 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 of Theorem 1.4 From the definition of strongly mixing we know that {Y k ,k ≥ 1} remain to be a sequence of identically distributed strongly mixing random variable with zero mean and unit variance. Let a k,n  b k,n /σ n ;notethat n  k1 b 2 k,n  b 1,n  2 n  k2 k−1  i1 1 k  b 1,n  2 n  k2 k − 1 k  2n − b 1,n ,n≥ 1, 2.10 and via 1.7 we have sup n n  k1 a 2 k,n  sup n n  k1 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 | 2δ } is uniformly integrable; note that inf k Var  Y k   EY 2 1  1 > 0, Var  n  k1 a k,n Y k   Var   n k1 b k,n Y k  σ 2 n  1, 2.12 Journal of Inequalities and Applications 5 and applying 1.5 ∞  n1 n 2/δ α  n   ∞  n1 n −r2/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  k1 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  k1 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  k1 d k  f  S k,k σ k  − Ef  S k,k σ k  a.s. −−−−→ 0,n−→ ∞. 2.18 Let ξ k  fS k,k /σ k  − EfS k,k /σ k , E  n  k1 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  k1 d k 2k  lk 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 2k1,l  S 2k   b 2k1,l Y 2k1  ··· 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 2k1,l  S 2k σ l             Cov  f  S k,k σ k  ,f  S l,l − S 2k,2k − b 2k1,l  S 2k σ l       , 2.23 and the properties of strongly mixing sequence imply      Cov  f  S k,k σ k  ,f  S l,l − S 2k,2k − b 2k1,l  S 2k σ l        α  k  . 2.24 Applying Lemma 2.3 and 2.10, Var  S 2k,2k   2k  i1 b 2 i,2k EY 2 i  2 2k−1  j1 2k  ij1 b i,2k b j,2k Cov  Y i ,Y j  ≤ 2k  i1 b 2 i,2k  2 2k−1  j1 b 2 j,2k 2k  ij1   Cov  Y i ,Y j     k, Var   S 2k   E  2k  i1 Y i  2  2k  i1 EY 2 i  2 2k−1  i1 2k  ji1 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 2k1,l  S 2k σ l        E    S 2k,2k  b 2k1,l  S 2k    σ l ≤  ES 2 2k,2k σ l   E  b 2k1,l  S 2k  2 σ l   Var  S 2k,2k  σ l  b 2k1,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  k1 n  l1 d k d l α  k    ln D n  −1−η n  k1 d k n  l1 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  k1 d k n  l1 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  k1 n 0  l2k d k d l ≤ e ln n α n  k1 d k n 0  l2k 1 l  e ln n α n  k1 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,and2.32, for some ε>0 such that ET 2 n  1 D 2 n E  n  k1 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 of Theorem 1.3 Let C k  S k /μk; we have 1 γσ n n  k1  C k − 1   1 γσ n n  k1  S k μk − 1   1 σ n n  k1 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  k1 d k I  1 γσ k k  i1  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  k1 d k I  1 γσ k k  i1 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 ln1  xx  Ox 2  for |x| < 1/2, thus      n  k1 ln  C k  − n  k1  C k − 1        n  k1  C k − 1  2  n  k1 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  k1  C k − 1  ≤ x − ε  ≤ I  1 γσ n n  k1 ln C k ≤ x  ≤ I  1 γσ n n  k1  C k − 1  ≤ x  ε  2.40 and thus 2.36 implies 2.37. 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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