Báo cáo hóa học: Research Article Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables

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Báo cáo hóa học: " Research Article Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables"Hindawi Publishing CorporationJournal of Inequalities and ApplicationsVolume 2011, Article ID 576301, 9 pagesdoi:10.1155/2011/576301Research ArticleAlmost Sure Central Limit Theorem for Product ofPartial 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 ResultsIn 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 Peligradand 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 xand 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 randomwhere Sn Nvariable e , and N is a standard normal variable. Recently, Jin 6 had proved that 1.1 holds under appropriate conditions for stronglymixing positive random variables and gave an ASCLT for product of partial sums of stronglymixing as follows.2 Journal of Inequalities and ApplicationsTheorem 1.2. Let {Xn , n ≥ 1} be a sequence of identically distributed positive strongly mixing n σ 2 , dkrandom variable with EX1 μ > 0 and Var X1 1/k, Dn k 1 dk . Denote by nγ 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 nThen for any real x ⎛ ⎞ 1/γσk k 1n i 1 Si dk I ⎝ ≤ x⎠ a.s. lim ...