On the asymptotic behaviour of increasing positive self-similar Markov processes

TL;DR

This paper investigates the asymptotic growth behavior of increasing positive self-similar Markov processes, establishing conditions for convergence and describing their growth rates using regular variation and law of iterated logarithm.

Contribution

It provides new results on the asymptotic behavior of ipssMp with infinite mean subordinator, including convergence criteria and growth rate characterizations.

Findings

Logarithm of ipssMp normalized by log time converges weakly under regular variation conditions.

Established a law of iterated logarithm for the liminf of the process's logarithm.

Derived an integral test for the upper envelope and analyzed the growth rate of the associated random clock.

Abstract

We are interested by the rate of growth of increasing positive self-similar Markov processes (ipssMp) such that the subordinator associated to it via Lamperti's transformation has infinite mean. We prove that the logarithm of an ipssMp normalized by the logarithm of the time converges weakly, as the time tends to infinity, if and only if the Laplace exponent of the underlying subordinator is regularly varying at zero. Moreover, we prove that the regular variation at zero of the Laplace exponent is essentially nasc for the existence of a function that normalizes the logarithm of an ipssMp. We obtain a law of iterated logarithm for the liminf of the logarithm of an ipssMp and an integral test to study the upper envelope of it. Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained.