On the rates of the other law of the logarithm

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 $a_n = oleft( {{{sqrt n } mathord{left/ {vphantom {{sqrt n } {log n}}} right. kern-nulldelimiterspace} {log n}}} right)$a_n = oleft( {{{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,

$mathop {lim }limits_{{{varepsilon nearrow 1} mathord{left/ {vphantom {{varepsilon nearrow 1} {sqrt {r - 1} }}} right. kern-nulldelimiterspace} {sqrt {r - 1} }}} left[ {varepsilon ^{ - 2} - left( {r - 1} right)} right]sumlimits_{n = 1}^infty {n^{r - 2} Pleft{ {M_n leqslant varepsilon sigma sqrt {{{pi ^2 n} mathord{left/ {vphantom {{pi ^2 n} {left( {8log n} right)}}} right. kern-nulldelimiterspace} {left( {8log n} right)}}} + a_n } right}} = frac{4} {pi }.$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]sumlimits_{n = 1}^infty {n^{r - 2} Pleft{ {M_n leqslant varepsilon sigma sqrt {{{pi ^2 n} mathord{left/ {vphantom {{pi ^2 n} {left( {8log n} right)}}} right. kern-nulldelimiterspace} {left( {8log 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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