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On the asymptotic behaviour of increasing self-similar Markov processes. (English) Zbl 1191.60047
Summary: It has been proved by J. Bertoin and M.-E. Caballero [Bernoulli 8, No. 2, 195–205 (2002; Zbl 1002.60032)] 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.

MSC:
60G18 Self-similar stochastic processes
60J25 Continuous-time Markov processes on general state spaces
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