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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 Cesàro 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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