Counting occurrences in almost sure limit theorems

Rita Giuliano-Antonini¹ ; Michel Weber²

¹ Dipartimento di Matematica Università di Pisa Via F. Buonarroti, 2 56127 Pisa, Italy
² Mathématique (IRMA) Université Louis-Pasteur 7, rue René Descartes 67084 Strasbourg Cedex, France

Colloquium Mathematicum, Tome 102 (2005) no. 2, pp. 271-290

Résumé

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.

Détail

DOI : 10.4064/cm102-2-8

Keywords: mathinner ldotp ldotp ldotp sequence random variables mathinner cdotp cdotp cdotp developing preceding work concerning case only compare under strictly weaker conditions those central limit theorem deviation series sum respect sum suitable weights arbitrary sequences reals extensions law belongs domain attraction p stable law obtained deduce strong versions central limit theorem