Quasi-stationary distributions and Yaglom limits for self-similar Markov processes
Vortrag von Dr. Victor Rivero
Sprecher eingeladen von: Prof. Dr. Jean Bertoin
Datum: 09.10.13 Zeit: 17.15 - 18.15 Raum: ETH
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 $\nu$. 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 discuss separately three cases, according whether the extinction time is in the domain of attraction of a Gumbel law, a Weibull law or a Fréchet law. In each of these cases, necessary and sufficient conditions on the parameters of the underlying Lévy 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 Fréchet 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 Lévy process. In that aim, some new results on its asymptotic behaviour are given. We will also discuss how the factorization of exponential, beta and Pareto distributions can be used to obtain explicit information about the law of the first hitting time of zero or of the Yaglom limit.