arXiv subjects

Yan-Xia Ren

Publications and source records attributed to Yan-Xia Ren.

59 records · Page 4Linked to original sources

arXiv2014

Small value probabilities for supercritical branching processes with immigration

We consider a supercritical Galton-Watson branching process with immigration. It is well known that under suitable conditions on the offspring and immigration distributions, there is a finite, strictly positive limit ${\mathcal{W}}$ for the normalized population size. Small value probabilities for ${\mathcal{W}}$ are obtained. Precise effects of the balance between offspring and immigration distributions are characterized.

math.PR↗

arXiv2013

Conditional limit theorems for critical continuous-state branching processes

In this paper we study the conditional limit theorems for critical continuous-state branching processes with branching mechanism $ψ(λ)=λ^{1+α}L(1/λ)$ where $α\in [0,1]$ and $L$ is slowly varying at $\infty$. We prove that if $α\in (0,1]$, there are norming constants $Q_{t}\to 0$ (as $t\uparrow +\infty$) such that for every $x>0$, $P_{x}\left(Q_{t}X_{t}\in\cdot|X_{t}>0\right)$ converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of $ψ$ at 0. We give a conditional limit theorem for the case $α=0$. The limit theorems we obtain in this paper allow infinite variance of the branching process.

math.PR↗

arXiv2013

Central Limit Theorems for Super-OU Processes

In this paper we study supercritical super-OU processes with general branching mechanisms satisfying a second moment condition. We establish central limit theorems for the super-OU processes. In the small and crtical branching rate cases, our central limit theorems sharpen the corresponding results in the recent preprint of Milos in that the limit normal random variables in our central limit theorems are non-degenerate. Our central limit theorems in the large branching rate case are completely new. The main tool of the paper is the so called "backbone decomposition" of superprocesses.

math.PR↗

arXiv2011

Strong law of large number of a class of super-diffusions

In this paper we prove that, under certain conditions, a strong law of large numbers holds for a class of super-diffusions $X$ corresponding to the evolution equation $\partial_t u_t=L u_t+βu_t-ψ(u_t)$ on a bounded domain $D$ in $\R^d$, where $L$ is the generator of the underlying diffusion and the branching mechanism $ψ(x,λ)=1/2α(x)λ^2+\int_0^\infty (e^{-λr}-1+λr)n(x, {\rm d}r)$ satisfies $\sup_{x\in D}\int_0^\infty (r\wedge r^2) n(x,{\rm d}r)<\infty$.

math.PR↗

arXiv2010

$L\log L$ condition for supercritical branching Hunt processes

In this paper we use the spine decomposition and martingale change of measure to establish a Kesten-Stigum $L\log L$ theorem for branching Hunt processes. This result is a generalization of the results in Asmussen-Hering (1976) and Hering (1978) for branching diffusions.

math.PR↗