The Self-Similar Dynamics of Renewal Processes

Albert Meads Fisher (University of Sao Paulo)
Marina Talet (Université de Provence)

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.

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Pages: 929-961

Publication Date: May 10, 2011

DOI: 10.1214/EJP.v16-888

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