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

TL;DR

This paper investigates the existence and characterization of quasi-stationary distributions and Yaglom limits for self-similar Markov processes reaching zero, classifying the limits based on the domain of attraction of the extinction time.

Contribution

It provides necessary and sufficient conditions for Yaglom limits in different attraction domains and links these limits to distribution factorizations, with new results on tail distributions of extinction times.

Findings

Yaglom limits exist iff extinction time is in a specific domain of attraction.

Conditions on Lévy process parameters determine the domain of attraction.

New results on tail distributions of extinction times are introduced.

Abstract

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 $\nu$ such that the process rescaled by $g$ and conditioned on non-extinction converges in distribution towards $\nu$. If the study of quasi-stationary distributions is easy and follows mainly from a previous result by Bertoin and Yor \cite{BYFacExp} and Berg \cite{bergI}, that of Yaglom limits is more challenging. We will see that a Yaglom limit exits 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 whether the extinction time is in the domain of attraction of a Gumbel law, a Weibull law or a Fr\'echet law. In each of these cases, necessary and sufficient conditions on the parameters of the underlying L\'evy 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 Fr\'echet case. This approach relies partly on results on the tail distribution of the extinction time, which is known to be distributed as the exponential integral of a L\'evy process. In that aim, new results on such tail distributions are given, which may be of independent interest. Last, we present applications of the Fr\'echet case to a family of Ornstein-Uhlenbeck processes.