Motivated by recent results on pathwise central limit theorems, we study in a systematic way log-average versions of classical limit theorems. For partial sums $S_k$ of independent r.v.'s we prove under mild technical conditions that $(1/\log N)\sum_{k \leq N}(1/k)I\{S_k/a_k \in \cdot\} \rightarrow G(\cdot)$ (a.s.) if and only if $(1/\log N)\sum_{k \leq N}(1/k)P(S_k/a_k \in \cdot) \rightarrow G(\cdot)$. A functional version of this result also holds. For partial sums of i.i.d. r.v.'s attracted to a stable law, we obtain a pathwise version of the stable limit theorem as well as a strong approximation by a stable process on log dense sets of integers. We also give necessary and sufficient conditions for the law of large numbers in log density.
Publié le : 1993-07-14
Classification:
Pathwise central limit theorem,
log-averaging methods,
stable convergence,
strong approximation,
law of large numbers,
60F05,
60F17,
60F15
@article{1176989135,
author = {Berkes, I. and Dehling, H.},
title = {Some Limit Theorems in Log Density},
journal = {Ann. Probab.},
volume = {21},
number = {4},
year = {1993},
pages = { 1640-1670},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989135}
}
Berkes, I.; Dehling, H. Some Limit Theorems in Log Density. Ann. Probab., Tome 21 (1993) no. 4, pp. 1640-1670. http://gdmltest.u-ga.fr/item/1176989135/