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November 2012 Random walks driven by low moment measures
Alexander Bendikov, Laurent Saloff-Coste
Ann. Probab. 40(6): 2539-2588 (November 2012). DOI: 10.1214/11-AOP687

Abstract

We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group $G$ and any positive function $\varrho\,\colon\,G\rightarrow [0,+\infty]$, we introduce a function $\Phi_{G,\varrho}$ which describes the fastest possible decay of $n\mapsto\phi^{(2n)}(e)$ when $\phi$ is a symmetric continuous probability density such that $\int\varrho\phi$ is finite. We estimate $\Phi_{G,\varrho}$ for a variety of groups $G$ and functions $\varrho$. When $\varrho$ is of the form $\varrho=\rho\circ\delta$ with $\rho\,\colon\,[0,+\infty)\rightarrow [0,+\infty)$, a fixed increasing function, and $\delta\,\colon\,G\rightarrow [0,+\infty)$, a natural word length measuring the distance to the identity element in $G$, $\Phi_{G,\varrho}$ can be thought of as a group invariant.

Citation

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Alexander Bendikov. Laurent Saloff-Coste. "Random walks driven by low moment measures." Ann. Probab. 40 (6) 2539 - 2588, November 2012. https://doi.org/10.1214/11-AOP687

Information

Published: November 2012
First available in Project Euclid: 26 October 2012

zbMATH: 1262.60005
MathSciNet: MR3050511
Digital Object Identifier: 10.1214/11-AOP687

Subjects:
Primary: 60B05 , 60J15

Keywords: group invariants , moments , Random walk

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 6 • November 2012
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