On the rates of the other law of the logarithm

Abstract

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

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

We also show that the widest a _n is $o\left( {{{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}} \right)$ .