On the asymptotic behaviour of increasing self-similar Markov processes
María Emilia Caballero, Instituto de Matemáticas UNAM Mexico
Víctor Manuel Rivero, Centro de Investigación en Matemáticas, Guanajuato
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 $ttoinfty,$ 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
$liminf_{ttoinfty}log(X(t))/log(t)=1/alpha,$ a.s. and provide
an integral test for the upper functions of ${log(X(t)), tgeq 0}.$ 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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