PRECISE ASYMPTOTICS IN LOGLOG LAW FOR ρ-MIXING RANDOM VARIABLES

• Journal title : Honam Mathematical Journal
• Volume 32, Issue 3,  2010, pp.525-536
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2010.32.3.525
Title & Authors
PRECISE ASYMPTOTICS IN LOGLOG LAW FOR ρ-MIXING RANDOM VARIABLES
Ryu, Dae-Hee;

Abstract
Let $\small{X_1,X_2,\cdots}$ be identically distributed $\small{\rho}$-mixing random variables with mean zeros and positive finite variances. In this paper, we prove $\small{\array{\lim\\{\in}\downarrow0}{\in}^2 \sum\limits_{n=3}^\infty\frac{1}{nlogn}P({\mid}S_n\mid\geq\in\sqrt{nloglogn}=1}$, $\small{\array{\lim\\{\in}\downarrow0}{\in}^2 \sum\limits_{n=3}^\infty\frac{1}{nlogn}P(M_n\geq\in\sqrt{nloglogn}=2 \sum\limits_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}}$ where $\small{S_n=X_1+\cdots+X_n,\;M_n=max_{1{\leq}k{\leq}n}{\mid}S_k{\mid}}$ and $\small{\sigma^2=EX_1^2+ 2\sum\limits{^{\infty}_{i=2}}E(X_1,X_i)=1}$.
Keywords
Precise asymptotics;Complete moment convergence;$\small{\rho}$mixing;Law of the iterated logarithm;
Language
English
Cited by
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