Let $\{X; X_1,X_2,... \}$ be a sequence of i.i.d. random variables with $X \in L^p$, $1 < p \leq 2$. For $n \geq 1$, let $S_n = X_1 + \dots+ X_n$. Developing a preceding work, adressing the $L^2$-case only, we compare, under strictly weaker conditions than those of the central limit theorem, the deviation of the series $\sum_n w_n 1_{S_n/\sqrt n<x_n}$ with respect to $\sum_n w_n P(S_n/\sqrt n<x_n)$, for suitable weights $(w_n)$ and arbitrary sequences $(s_n)$ of reals . Extensions to the case $1 < p < 2$, with suitable norming constants, and when the law of $X$ belongs to the domain of attraction of a $p$-stable law, are obtained. We deduce strong versions of the a.s. central limit theorem.
Counting occurrences in almost sure limit theorems, "Colloquium Mathematicum", 102 (2), 271-290 (with M. Weber)
GIULIANO, RITA;
2005-01-01
Abstract
Let $\{X; X_1,X_2,... \}$ be a sequence of i.i.d. random variables with $X \in L^p$, $1 < p \leq 2$. For $n \geq 1$, let $S_n = X_1 + \dots+ X_n$. Developing a preceding work, adressing 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 1_{S_n/\sqrt nI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.