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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence | 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 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 Xn n 1 be a standardized non-stationary Gaussian sequence and let denote Sn ỵn 1 Xk an yfVar Sn . Under some additional condition let the constants uni 1 i n n 1 satisfy 2n 1 1 - uni T as n OT for some T 0 and min1 i n uni c logn 1 2 for some c 0 then we have limn 0 1 log n 2k 1 1 k I ni 1 Xi Uki Sk ơk x e-T x almost surely for any x e R where I A is the indicator function of the event A and x stands for the standard normal distribution function. 1. Introduction When X Xn n 1 is a sequence of independent and identically distributed i.i.d. random variables and Sn n 1 Xk n 1 Mn max1 k nXk 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 -X log n xi x vk 1.1 almost surely for all x e R where I A is the indicator function of the event A and 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 Stadtmuller 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 .
