Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

Authors:	Mi Chenjing

Institution:	(1) Dept. of Math, Zhejiang Univ., 310028 Hangzhou, China

Abstract:	Let {X _i;i≥1} be a strictly stationary sequence of associated random variables with mean zero and let $\sigma ^2 = EX_1^2 + 2\sum\nolimits_{j = 2}^\infty {EX_1 X_j }$ with O<σ ²<∞. Set $S_n = \sum\nolimits_{i = 1}^n {X_i }$ , the precise asymptotics for $\sum\nolimits_{n \geqslant 1} {n^{\frac{r}{p} - 2} } P\left( {\left\| {S_n } \right\| \geqslant \varepsilon n^{\frac{1}{p}} } \right),\sum\nolimits_{n \geqslant 1} {\frac{1}{n}P} \left( {\left\| {S_n } \right\| \geqslant \varepsilon n^{\frac{1}{p}} } \right)$ and $\sum\nolimits_{n \geqslant 1} {\frac{{\left( {log n} \right)^\delta }}{n}} P\left( {\left\| {S_n } \right\| \geqslant \varepsilon \sqrt {nlogn} } \right)$ as $\varepsilon \searrow 0$ are established.

Keywords:	complete convergence associated random variables Baum-Katz law precise asymptotics
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