Stationary measures and the continuous-state branching process conditioned on extinction

TL;DR

This work analyzes stationary measures for continuous-state branching processes that become extinct almost surely by connecting the stationary measure to the scale function of the associated Lévy process. It shows that the nontrivial stationary measure on $(0,\infty)$ is given by $\mu(dx)=\frac{W(x)}{x}\,dx$ (up to a constant) and can be obtained as a vague limit of the potential measure; in the critical case it also emerges from a suitably normalized transition semigroup. The paper then develops conditional limit theorems for extinction in two regimes—near extinction and extinction at a fixed time—revealing size-biased versions of the stationary measure in the critical case and Yaglom-type limits in the subcritical case, with explicit Laplace-transform characterizations. It also provides detailed descriptions of the limiting distributions, their infinite-divisibility, and asymptotic connections to the Q-process. Overall, the results deepen the understanding of long-run and conditional behavior for extinction-prone CB processes via scale-function methods and Laplace-transform techniques.

Abstract

We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom's distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

Theorems & Definitions (34)

• Theorem 2.1
• proof
• Remark 2.1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Example 3.3
• Example 3.4
• Theorem 3.5