On the LIL for Self-Normalized Sums of IID Random Variables

Abstract

Let $X,X_i ,i \in \mathbb{N},$ be i.i.d. random variables and let, for each $n \in \mathbb{N},S_n = \sum\nolimits_{i = 1}^n {X_i }$ and $V_n^2 = \sum\nolimits_{i = 1}^n {X_i^2 }$. It is shown that $\lim \sup _{n \to \infty } {{|S_n |} \mathord{\left/ {\vphantom {{|S_n |} {(V_n \sqrt {\log \log n} ) < \infty }}} \right. \kern-\nulldelimiterspace} {(V_n \sqrt {\log \log n} ) < \infty }}$ a.s. whenever the sequence of self-normalized sums S _n /V _n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if $\lim \sup _{t \to \infty } {{[t|\mathbb{E}XI(|X| \leqslant t)|} \mathord{\left/ {\vphantom {{[t|\mathbb{E}XI(|X| \leqslant t)|} {\mathbb{E}X^2 I(|X| \leqslant t)] < \infty }}} \right. \kern-\nulldelimiterspace} {\mathbb{E}X^2 I(|X| \leqslant t)] < \infty }}$