## The Annals of Probability

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

### Some Limit Theorems in Log Density

I. Berkes and H. Dehling

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989135

**Digital Object Identifier**

doi:10.1214/aop/1176989135

**Mathematical Reviews number (MathSciNet)**

MR1235433

**Zentralblatt MATH identifier**

0785.60014

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60F17: Functional limit theorems; invariance principles 60F15: Strong theorems

**Keywords**

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

#### Citation

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