Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct
Amaury Lambert, University Paris 6
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 Llog 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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