@article{EJP888,
author = {Albert Fisher and Marina Talet},
title = {The Self-Similar Dynamics of Renewal Processes},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {16},
year = {2011},
keywords = {stable process, renewal process, Mittag-Leffler process, Cauchy process, almost-sure invariance principle in log density, pathwise Central Limit Theorem.},
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.},
pages = {no. 31, 929-961},
issn = {1083-6489},
doi = {10.1214/EJP.v16-888},
url = {http://ejp.ejpecp.org/article/view/888}}