Quasi-stationary distributions and Yaglom limits of self-similar Markov processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:122

Article

Haas, Benedicte^1,2 

[1] Univ Paris 09, F-75775 Paris 16, France

[2] Ecole Normale Super, Paris, France

关键词: Self-similar Markov processes;    Levy processes;    Yaglom limits;    Quasi-stationary distributions;    Extreme value theory;    Exponential functionals of Levy processes;   

DOI  :  10.1016/j.spa.2012.08.006

来源: Elsevier

【 摘 要 】

We discuss the existence and characterization of quasi-stationary distributions and Yaglom limits of self-similar Markov processes that reach 0 in finite time. By Yaglom limit, we mean the existence of a deterministic function g and a non-trivial probability measure v such that the process rescaled by g and conditioned on non-extinction converges in distribution towards v. We will see that a Yaglom limit exists if and only if the extinction time at 0 of the process is in the domain of attraction of an extreme law and we will then treat separately three cases, according to whether the extinction time is in the domain of attraction of a Gumbel, Weibull or Frechet law. In each of these cases, necessary and sufficient conditions on the parameters of the underlying Levy process are given for the extinction time to be in the required domain of attraction. The limit of the process conditioned to be positive is then characterized by a multiplicative equation which is connected to a factorization of the exponential distribution in the Gumbel case, a factorization of a Beta distribution in the Weibull case and a factorization of a Pareto distribution in the Frechet case. This approach relies partly on results on the tail distribution of the extinction time, which is known to be distributed as the exponential functional of a Levy process. In that aim, new results on such tail distributions are given, which may be of independent interest. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free