Some Limit Theorems in Log Density

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.