Asymptotic properties for the loglog laws under positive association

Mathematica Slovaca

Xiao-Rong Yang¹  Ke-Ang Fu¹ 

关键词: the law of the iterated logarithm;    Chung-type law of the iterated logarithm;    positive association;    moment convergence;    tail probability;   

DOI  :  10.2478/s12175-012-0059-0

学科分类：数学（综合）

来源: Slovenska Akademia Vied * Matematicky Ustav / Slovak Academy of Sciences, Mathematical Institute

【 摘 要 】

Let {X n: n ≥ 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $$S_n = sumlimits_{k = 1}^n {X_k }$$, $$Mn = mathop {max }limits_{k leqslant n} left| {S_k } ight|$$, n ≥ 1. Suppose that $$0 < sigma ^2 = EX_1^2 + 2sumlimits_{k = 2}^infty {EX_1 X_k < infty }$$. In this paper, we prove that if E|X 1|2+δ < for some δ ∈ (0, 1], and $$sumlimits_{j = n + 1}^infty {Covleft( {X_1 ,X_j } ight) = Oleft( {n^{ - alpha } } ight)}$$ for some α > 1, then for any b > −1/2 $$mathop {lim }limits_{varepsilon searrow 0} varepsilon ^{2b + 1} sumlimits_{n = 1}^infty {frac{{(log log n)^{b - 1/2} }}{{n^{3/2} log n}}} Eleft{ {M_n - sigma varepsilon sqrt {2nlog log n} } ight}_ + = frac{{2^{ - 1/2 - b} Eleft| N ight|^{2(b + 1)} }}{{(b + 1)(2b + 1)}}sumlimits_{k = 0}^infty {frac{{( - 1)^k }}{{(2k + 1)^{2(b + 1)} }}}$$ and $$mathop {lim }limits_{varepsilon earrow infty } varepsilon ^{ - 2(b + 1)} sumlimits_{n = 1}^infty {frac{{(log log n)^b }}{{n^{3/2} log n}}Eleft{ {sigma varepsilon sqrt {frac{{pi ^2 n}}{{8log log n}}} - M_n } ight}} _ + = frac{{Gamma (b + 1/2)}}{{sqrt 2 (b + 1)}}sumlimits_{k = 0}^infty {frac{{( - 1)^k }}{{(2k + 1)^{2b + 2} }}} ,$$ where x + = max{x, 0}, N is a standard normal random variable, and Γ(·) is a Gamma function.

【 授权许可】

Unknown