Electronic Journal of Probability

Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct

Amaury Lambert

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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.

Article information

Electron. J. Probab., Volume 12 (2007), paper no. 14, 420-446.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K05: Renewal theory 60F05: Central limit and other weak theorems 60H10: Stochastic ordinary differential equations [See also 34F05] 60G18: Self-similar processes

Continuous-state branching process Lévy process quasi-stationary distribution Yaglom theorem h-transform Q-process immigration size-biased distribution stochastic differential equations

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Lambert, Amaury. Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct. Electron. J. Probab. 12 (2007), paper no. 14, 420--446. doi:10.1214/EJP.v12-402. https://projecteuclid.org/euclid.ejp/1464818485

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