The Self-Similar Dynamics of Renewal Processes
Résumé
We prove an almost sure invariance principle in log density for renewal processes with gaps in the domain of attraction of an α
-stable law. There are three different types of behavior: attraction to a Mittag-Leffler process for 0<α<1, to a centered Cauchy process for α=1 and to a stable process for 1<α≤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.