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On the rates of the other law of the logarithm
Authors:Li-Xin Zhang and You-You Chen
Institution:(1) Department of Mathematics, Zhejiang University, Hangzhou, 310028, P. R. China
Abstract:Let X, X 1, X 2, … be i.i.d. random variables, and set S n = X 1 + … + X n , M n = max kn |S k |, n ≧ 1. Let $a_n = o\left( {{{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}} \right)$a_n = o\left( {{{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}} \right) . By using the strong approximation, we prove that, if EX = 0, VarX = σ 2 > 0 and E|X|2+ε < ∞ for some ε > 0, then for any r > 1,
$\mathop {\lim }\limits_{{{\varepsilon \nearrow 1} \mathord{\left/ {\vphantom {{\varepsilon \nearrow 1} {\sqrt {r - 1} }}} \right. \kern-\nulldelimiterspace} {\sqrt {r - 1} }}} \left {\varepsilon ^{ - 2} - \left( {r - 1} \right)} \right]\sum\limits_{n = 1}^\infty {n^{r - 2} P\left\{ {M_n \leqslant \varepsilon \sigma \sqrt {{{\pi ^2 n} \mathord{\left/ {\vphantom {{\pi ^2 n} {\left( {8\log n} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {8\log n} \right)}}} + a_n } \right\}} = \frac{4} {\pi }.$\mathop {\lim }\limits_{{{\varepsilon \nearrow 1} \mathord{\left/ {\vphantom {{\varepsilon \nearrow 1} {\sqrt {r - 1} }}} \right. \kern-\nulldelimiterspace} {\sqrt {r - 1} }}} \left {\varepsilon ^{ - 2} - \left( {r - 1} \right)} \right]\sum\limits_{n = 1}^\infty {n^{r - 2} P\left\{ {M_n \leqslant \varepsilon \sigma \sqrt {{{\pi ^2 n} \mathord{\left/ {\vphantom {{\pi ^2 n} {\left( {8\log n} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {8\log n} \right)}}} + a_n } \right\}} = \frac{4} {\pi }.
Keywords:Complete convergence  tail probabilities of sums of i  i  d  random variables  the other lawof the logarithm  strong approximation
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