On the rates of the other law of the logarithm |
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Authors: | Li-Xin Zhang and You-You Chen |
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Institution: | (1) Department of Mathematics, Zhejiang University, Hangzhou, 310028, P. R. China |
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Abstract: | Let X, X
1, X
2, … be i.i.d. random variables, and set S
n
= X
1 + … + X
n
, M
n
= max
k≤n
|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 }. |
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Keywords: | Complete convergence tail probabilities of sums of i i d random variables the other lawof the logarithm strong approximation |
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