MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES

Title & Authors
MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES
Fu, Ke-Ang; Hu, Li-Hua;

Abstract
Let {$\small{X_n;n\;\geq\;1}$} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $\small{S_n\;=\;{\sum}^n_{k=1}X_k}$, $\small{M_n\;=\;max_{k{\leq}n}|S_k|}$, $\small{n\;{\geq}\;1}$. Suppose $\small{\sigma^2\;=\;EX^2_1+2{\sum}^\infty_{k=2}EX_1X_k}$ (0 < $\small{\sigma}$ < $\small{\infty}$). We prove that for any b > -1/2, if $\small{E|X|^{2+\delta}}$(0<$\small{\delta}$$\small{\leq}$1), then $\small{lim\limits_{\varepsilon\searrow0}\varepsilon^{2b+1}\sum^{\infty}_{n=1}\frac{(loglogn)^{b-1/2}}{n^{3/2}logn}E\{M_n-\sigma\varepsilon\sqrt{2nloglogn}\}_+=\frac{2^{-1/2-b}{\sigma}E|N|^{2(b+1)}}{(b+1)(2b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2(b+1)}}}$ and for any b > -1/2, $\small{lim\limits_{\varepsilon\nearrow\infty}\varepsilon^{-2(b+1)}\sum^{\infty}_{n=1}\frac{(loglogn)^b}{n^{3/2}logn}E\{\sigma\varepsilon\sqrt{\frac{\pi^2n}{8loglogn}}-M_n\}_+=\frac{\Gamma(b+1/2)}{\sqrt{2}(b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2b+2$, where $\small{\Gamma(\cdot)}$ 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;
Language
English
Cited by
1.
Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of φ-Mixing Sequences, Mathematica Slovaca, 2015, 65, 6
2.
Precise rates of the first moment convergence in the LIL for NA sequences, Mathematische Nachrichten, 2014, 287, 17-18, 2138
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