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The self-similar dynamics of renewal processes. (English) Zbl 1225.60143
Summary: 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 CesĂ ro density zero, in the stable manifold of a self-similar path for the scaling flow. As a corollary, we obtain pathwise functional and central limit theorems.

MSC:
60K05 Renewal theory
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60F17 Functional limit theorems; invariance principles
60G18 Self-similar stochastic processes
60G52 Stable stochastic processes
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