Counting occurrences in almost sure limit theorems
Volume 102 / 2005
Colloquium Mathematicum 102 (2005), 271-290
MSC: Primary 60F15, 60G50; Secondary 60F05, 60E07.
DOI: 10.4064/cm102-2-8
Abstract
Let $X,X_1,X_2,\mathinner {\ldotp \ldotp \ldotp }$ be a sequence of i.i.d. random variables with $X\in L^p$, $0< p\le 2$. For $n\ge 1$, let $S_n= X_1+\mathinner {\cdotp \cdotp \cdotp }+ X_n$. Developing a preceding work concerning the $L^2$-case only, we compare, under strictly weaker conditions than those of the central limit theorem, the deviation of the series $\sum _n w_n{\bf 1}_{\{ S_n < s_n\} } $ with respect to $\sum _n w_n{\bf P}\{ S_n < s_n\} $, for suitable weights $(w_n)$ and arbitrary sequences $(s_n)$ of reals. Extensions to the case $0< p< 2$, and when the law of $X$ belongs to the domain of attraction of a $p$-stable law, are also obtained. We deduce strong versions of the a.s. central limit theorem.
