The Self-Similar Dynamics of Renewal Processes

Abstract

We prove an almost sure invariance principle in log density for renewal processes with gaps in the domain of attraction of an $\alpha$-stable law. There are three different types of behavior: attraction to a Mittag-Leffler process for $0<\alpha<1$, to a centered Cauchy process for $\alpha=1$ and to a stable process for $1<\alpha\leq 2$. Equivalently, in dynamical terms, almost every renewal path is, upon centering and up to a regularly varying coordinate change of order one, and after removing a set of times of Cesàro density zero, in the stable manifold of a self-similar path for the scaling flow. As a corollary we have pathwise functional and central limit theorems.