Functional laws of the Iterated Logarithm for the Partial Sums of I. I. D. Random Variables in the Domain of Attraction of a Completely Asymmetric Stable Law

Abstract

Suppose $X$ and $X_n, n \geqq 1$, are i.i.d. random variables whose common distribution lies in the domain of attraction of a completely asymmetric stable law of index $\alpha (0 < \alpha < 2)$, so that (i) as $\nu \rightarrow \infty, \nu \rightarrow P\{X \geqq \nu\}$ varies regularly with exponent $-\alpha$, and (ii) $\lim_{\nu\rightarrow\infty} P\{X \leqq - \nu\}/P\{X \geqq \nu\} = 0$. Under a condition only slightly more strigent than (ii), we present Strassen-type functional laws of the iterated logarithm for the partial sums $S_n = \sum_{m\leqq n} X_m, n \geqq 1$. Our laws hold in particular when $X \geqq 0$; the proofs in this case utilize some new large deviation results for the $S_n$'s.