The Asymptotic Behavior for Generalized Jiřina Process

As the classic branching process, the Galton-Watson process has obtained intensive attentions in the past decades. However, this model has two idealized assumptions–discrete states and time-homogeneity. In the present paper, we consider a branching process with continuous states, and for any given $n\in \mathbb{N}$, the branching law of every particle in generation n is determined by the population size of generation $n.$ We consider the case that the process is extinct with Probability 1 since in this case the process will be substantially different from the size-dependent branching process with discrete states. We give the extinction rate in the sense of $L^2$ and almost surely by the form of harmonic moments, that is to say, we show how fast $\left\{Z_n^{-1}\right\}$ grows under a group of sufficient conditions. From the result of the present paper, we observe that the extinction rate will be determined by an asymptotic behavior of the mean of the branching law. The results obtained in this paper have the more superiority than the counterpart from the existing literature.