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Precise asymptotics in the law of the iterated logarithm*

Authors:

Yuexu Zhao

Affiliation:

(1) Dept. of Infor. and Math. Sci., Hangzhou Dianzi University, Hangzhou, 310018, CHINA

Abstract:

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and $varepsilon downarrow {sqrt {alpha + 1} },EX^{2}_{1} {left( {log {left| {X_{1} } right|}} right)}^{{alpha + 1}} {left( {log {kern 1pt};log {left| {X_{1} } right|}} right)}^{{beta + 1}} < infty$ , we prove that

$begin{aligned} & {mathop {lim }limits_{varepsilon downarrow {sqrt {alpha + 1} }} }{left( {varepsilon ^{2} - {left( {alpha + 1} right)}} right)}^{{beta + 1/2}} {sumlimits_{n geqslant 3} {frac{{{left( {log n} right)}^{alpha } {left( {log log n} right)}beta }} {n}} } & P{left( {{left| {S_{n} } right|} geqslant sigma {sqrt {2nlog log n} }{left( {varepsilon + kappa _{n} {left( varepsilon right)}} right)}} right)} & = {left( {1/{sqrt pi }} right)}{left( {alpha + 1} right)}^{{ - 1/2}} exp {left( { - 2lambda {sqrt {alpha + 1} }} right)}Gamma {left( {beta + 1/2} right)}. end{aligned}$

*Supported by the Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. 20060237 and 20050494).

Keywords:

KeywordHeading" >: precise asymptotics the law of the iterated logarithm partial sums

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