The Annals of Probability

Some Limit Theorems in Log Density

I. Berkes and H. Dehling

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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.

Article information

Ann. Probab., Volume 21, Number 3 (1993), 1640-1670.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles 60F15: Strong theorems

Pathwise central limit theorem log-averaging methods stable convergence strong approximation law of large numbers


Berkes, I.; Dehling, H. Some Limit Theorems in Log Density. Ann. Probab. 21 (1993), no. 3, 1640--1670. doi:10.1214/aop/1176989135.

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