Precise asymptotics in the law of the iterated logarithm*

Let X1, X2, ... be i.i.d. random variables with EX1 = 0 and positive, finite variance σ2, and set Sn = X1 + ... + Xn. For any α > −1, β > −1/2 and for κn(ε) a function of ε and n such that κn(ε) log log n → λ as n ↑ ∞ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varepsilon \downarrow {\sqrt {\alpha + 1} },EX^{2}_{1} {\left( {\log {\left| {X_{1} } \right|}} \right)}^{{\alpha + 1}} {\left( {\log {\kern 1pt}\;\log {\left| {X_{1} } \right|}} \right)}^{{\beta + 1}} < \infty $$\end{document}, we prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} & {\mathop {\lim }\limits_{\varepsilon \downarrow {\sqrt {\alpha + 1} }} }{\left( {\varepsilon ^{2} - {\left( {\alpha + 1} \right)}} \right)}^{{\beta + 1/2}} {\sum\limits_{n \geqslant 3} {\frac{{{\left( {\log n} \right)}^{\alpha } {\left( {\log \log n} \right)}\beta }} {n}} } \\ & P{\left( {{\left| {S_{n} } \right|} \geqslant \sigma {\sqrt {2n\log \log n} }{\left( {\varepsilon + \kappa _{n} {\left( \varepsilon \right)}} \right)}} \right)} \\ & = {\left( {1/{\sqrt \pi }} \right)}{\left( {\alpha + 1} \right)}^{{ - 1/2}} \exp {\left( { - 2\lambda {\sqrt {\alpha + 1} }} \right)}\Gamma {\left( {\beta + 1/2} \right)}. \\ \end{aligned} $$\end{document}