General One-Sided Laws of the Iterated Logarithm

Abstract

Let $\{X_i\}$ be a sequence of independent, identically distributed nondegenerate random variables and $S_n = \sum^n_{i = 1}X_i$. We consider the question for various centering sequences $\{\alpha_n\}$: when is it possible to find a positive, monotone sequence $\{\beta_n\}$ such that $\lim \sup \beta^{-1}_n (S_n - \alpha_n) = c$ a.s., $c$ a finite nonzero constant? If $\alpha_n = \operatorname{med} S_n$, we obtain a necessary and sufficient condition for this. An important corollary is a one-sided version of the Hartman-Wintner law of the iterated logarithm: if $E(X^+)^2 < \infty$, then it is always possible to find such a norming sequence. Explicit norming sequences are given which are easy to obtain. Necessary and sufficient conditions are also given for being able to find a norming sequence $\{\beta_n\}$ for the two-sided problem $(\lim \sup \beta^{-1}_n |S_n - \alpha_n| = c$ a.s.) when $\alpha_n = ES_n$ and $\alpha_n = 0$. The two-sided problem with $\alpha_n = \operatorname{med} S_n$ was solved by Kesten. The one-sided problem remains open for $\alpha_n = ES_n$ and $\alpha_n = 0$. Examples are given which illustrate the advantage of considering different centering sequences. A one-sided version of Strassen's converse to the law of the iterated logarithm is also given: if $\lim \sup S_n/ \sqrt{2n \log \log n} = 1$ a.s., then $EX = 0, EX^2 = 1$.