• Moment Convergence Rates in the Law of the Logarithm for Dependent Sequences

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/119/03/0387-0400

• Keywords

The law of the logarithm; Chung-type law of the logarithm; negative association; moment convergence; tail probability.

• Abstract

Let $\{X_n;n\geq 1\}$ be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $S_n=\sum^n_{k=1}X_k,M_n=\max_{k\leq n}|S_k|,n\geq 1$. Suppose $\sigma^2=EX^2_1+2\sum^\infty_{k=2}EX_1X_k(0 &lt; \sigma &lt; \infty)$. In this paper, the exact convergence rates of a kind of weighted infinite series of $E\{M_n-\sigma\varepsilon\sqrt{n\log n}\}_+$ and $E\{|S_n|-\sigma\varepsilon\sqrt{n\log n}\}_+$ as $\varepsilon\searrow 0$ and $E\{\sigma\varepsilon\sqrt{\frac{\pi^2 n}{8\log n}}-M_n\}_+$ as $\varepsilon\nearrow\infty$ are obtained.

• Author Affiliations

1. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310 018, China

• Proceedings – Mathematical Sciences

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November 2019

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