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2009 On the asymptotic behaviour of increasing self-similar Markov processes
María Caballero, Víctor Rivero
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Electron. J. Probab. 14: 865-894 (2009). DOI: 10.1214/EJP.v14-637

Abstract

It has been proved by Bertoin and Caballero cite{BC2002} that a $1/\alpha$-increasing self-similar Markov process $X$ is such that $t^{-1/\alpha}X(t)$ converges weakly, as $t\to\infty,$ to a degenerate random variable whenever the subordinator associated to it via Lamperti's transformation has infinite mean. Here we prove that $\log(X(t)/t^{1/\alpha})/\log(t)$ converges in law to a non-degenerate random variable if and only if the associated subordinator has Laplace exponent that varies regularly at $0.$ Moreover, we show that $\liminf_{t\to\infty}\log(X(t))/\log(t)=1/\alpha,$ a.s. and provide an integral test for the upper functions of $\{\log(X(t)), t\geq 0\}.$ Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained. In particular, these allow us to establish estimates for the left tail of some exponential functionals of subordinators. Finally, some of the implications of these results in the theory of self-similar fragmentations are discussed.

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María Caballero. Víctor Rivero. "On the asymptotic behaviour of increasing self-similar Markov processes." Electron. J. Probab. 14 865 - 894, 2009. https://doi.org/10.1214/EJP.v14-637

Information

Accepted: 19 April 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1191.60047
MathSciNet: MR2497455
Digital Object Identifier: 10.1214/EJP.v14-637

Subjects:
Primary: 60G18
Secondary: 60B10, 60G17

JOURNAL ARTICLE
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