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


 
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1. Title Title of document Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct
 
2. Creator Author's name, affiliation, country Amaury Lambert; University Paris 6
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Continuous-state branching process; Lévy process; quasi-stationary distribution; Yaglom theorem; h-transform; Q-process; immigration; size-biased distribution; stochastic differential equations
 
3. Subject Subject classification 60J80; 60K05; 60F05; 60H10; 60G18
 
4. Description Abstract 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) CNRS
 
7. Date (YYYY-MM-DD) 2007-04-07
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/402
 
10. Identifier Digital Object Identifier 10.1214/EJP.v12-402
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 12
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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