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2011 The Self-Similar Dynamics of Renewal Processes
Albert Fisher, Marina Talet
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Electron. J. Probab. 16: 929-961 (2011). DOI: 10.1214/EJP.v16-888


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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Albert Fisher. Marina Talet. "The Self-Similar Dynamics of Renewal Processes." Electron. J. Probab. 16 929 - 961, 2011.


Accepted: 10 May 2011; Published: 2011
First available in Project Euclid: 1 June 2016

zbMATH: 1225.60143
MathSciNet: MR2801456
Digital Object Identifier: 10.1214/EJP.v16-888

Primary: 37A50, 60F17, 60K05
Secondary: 0G52, 60G18


Vol.16 • 2011
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