Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2010; 47(2): 263-275

Printed March 1, 2010

https://doi.org/10.4134/JKMS.2010.47.2.263

Copyright © The Korean Mathematical Society.

Moment convergence rates of LIL for negatively associated sequences

Ke-Ang Fu and Li-Hua Hu

Zhejiang Gongshang University and Zhejiang Gongshang University

Abstract

Let $\{X_n;n\geq1\}$ be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $S_n=\sum_{k=1}^{n}X_k$, $M_n=\max_{k\leq n}|S_k| , n \geq 1$. Suppose $\sigma^2=\boldsymbol{\rm E} X_1^2+2\sum_{k=2}^{\infty}\boldsymbol{\rm E} X_1X_k$ $(0<\sigma <\infty)$. We prove that for any $b> - 1/2$, if $\boldsymbol{\rm E}|X|^{2+\delta} (0<\delta\le 1)$, then $$\begin{aligned} &\ \lim\limits_{\varepsilon\searrow 0}\varepsilon^{2b+1}\sum\limits_{n=1}^{\infty}\frac{(\log \log n)^{b- 1/2}}{n^{3/2}\log n} \boldsymbol{E}\bigg\{M_n - \sigma\varepsilon\sqrt{2n\log \log n}\bigg\}_+\\=&\ \frac{2^{-1/2-b}\sigma\boldsymbol{\rm E}|N|^{2(b+1)}}{(b+1)(2b+1)}\sum\limits_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2(b+1)}}, \end{aligned}$$
and for any $b>-1/2,$ $$\begin{aligned} &\ \lim\limits_{\varepsilon\nearrow\infty}\varepsilon^{-2(b+1)}\sum\limits_{n=1}^{\infty}\frac{(\log \log n)^b}{n^{3/2}\log n} \boldsymbol{\rm E}\bigg\{\sigma\varepsilon\sqrt{\frac{\pi^2n}{8\log\log n}}-M_n\bigg\}_+\\=&\ \frac{\Gamma(b+1/2)}{\sqrt{2}(b+1)}\sum\limits_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2b+2}}, \end{aligned}$$ where $\Gamma(\cdot)$ is the Gamma function and $N$ stands for the standard normal random variable.

Keywords: Chung-type law of the iterated logarithm, moment convergence rates, negative association, the law of the iterated logarithm

MSC numbers: 60F15, 60G50