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2007 Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct
Amaury Lambert
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Electron. J. Probab. 12: 420-446 (2007). DOI: 10.1214/EJP.v12-402


We consider continuous-state branching (CB) processes which become extinct (i.e., hit 0) with positive probability. We characterize all the quasi-stationary distributions (QSD) for the CB-process as a stochastically monotone family indexed by a real number. We prove that the minimal element of this family is the so-called Yaglom quasi-stationary distribution, that is, the limit of one-dimensional marginals conditioned on being nonzero. Next, we consider the branching process conditioned on not being extinct in the distant future, or $Q$-process, defined by means of Doob $h$-transforms. We show that the $Q$-process is distributed as the initial CB-process with independent immigration, and that under the $L\log L$ condition, it has a limiting law which is the size-biased Yaglom distribution (of the CB-process). More generally, we prove that for a wide class of nonnegative Markov processes absorbed at 0 with probability 1, the Yaglom distribution is always stochastically dominated by the stationary probability of the $Q$-process, assuming that both exist. Finally, in the diffusion case and in the stable case, the $Q$-process solves a SDE with a drift term that can be seen as the instantaneous immigration.


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Amaury Lambert. "Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct." Electron. J. Probab. 12 420 - 446, 2007.


Accepted: 7 April 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1127.60082
MathSciNet: MR2299923
Digital Object Identifier: 10.1214/EJP.v12-402

Primary: 60J80
Secondary: 60F05 , 60G18 , 60H10 , 60K05

Keywords: Continuous-state branching process , H-transform , immigration , Lévy process , Q-process , quasi-stationary distribution , Size-biased distribution , Stochastic differential equations , Yaglom theorem

Vol.12 • 2007
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