Precise asymptotics in Chung’s law of the iterated logarithm

Abstract

Let X, X ₁, X ₂, ... be i.i.d. random variables with mean zero and positive, finite variance σ ², and set S _n = X ₁ + ... + X _n , n ≥ 1. The author proves that, if EX ² I{|X| ≥ t} = o((log log t)⁻¹) as t → ∞, then for any a > −1 and b > −1,

$$ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \nearrow 1/\sqrt {1 + a} } \left( {\frac{1} {{\sqrt {1 + a} }} - \varepsilon } \right)^{b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log n)^a (\log \log n)^b }} {n}P} \left\{ {\mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right| \leqslant \sqrt {\frac{{\sigma ^2 \pi ^2 n}} {{8\log \log n}}} (\varepsilon + a_n )} \right\} \hfill \\ = \frac{4} {\pi }\left( {\frac{1} {{2(1 + a)^{3/2} }}} \right)^{b + 1} \Gamma (b + 1) \hfill \\ \end{gathered} $$

, whenever a _n = o(1/ log log n). The author obtains the sufficient and necessary conditions for this kind of results to hold.