Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Abstract

Let {X,X _n ; n ≥ 1} be a sequence of i.i.d. random variables, EX = 0, EX ² = σ ² < ∞. Set S _n = X ₁ + X ₂ + ⋯ + X _n , M _n = max _k≤n ∣S _k ∣, n ≥ 1. Let a _n = O(1/ log log n). In this paper, we prove that, for b > −1,

$$ \begin{aligned} & {\mathop {\lim }\limits_{\varepsilon \searrow 0} }\varepsilon ^{{2{\left( {b + 1} \right)}}} {\sum\limits_{n = 1}^\infty {\frac{{{\left( {\log \;\log \;n} \right)}^{b} }} {{n\log n}}} }n^{{ - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\rm E}{\left\{ {M_{n} - \sigma {\left( {\varepsilon + a_{n} } \right)}{\sqrt {2n\log \;\log \;n} }} \right\}} + \\ & = \frac{{\sigma 2^{{ - b}} }} {{{\left( {b + 1} \right)}{\left( {2b + 3} \right)}}}{\rm E}{\left| N \right|}^{{2b + 3}} {\sum\limits_{k = 0}^\infty {\frac{{{\left( { - 1} \right)}^{k} }} {{{\left( {2k + 1} \right)}^{{2b + 3}} }}} } \\ \end{aligned} $$

holds if and only if EX = 0 and EX ² = σ ² < ∞.