Null recurrent birth-and-death processes, limits of certain martingales, and local asymptotic mixed normality
研究了生命周期长度尾部指数为a（0<a<1）的零常返生灭过程，证明了某些鞅弱收敛于时间变换的布朗运动，并应用于临界带移民分支过程的局部渐近混合正态性及参数估计比较。
We consider recurrent birth-and-death processes where the tails of the lifecycle length distribution vary regularly at X with index -a, O<a<1. We prove weak convergence of certain martingales (essentially compensated counting processes) to time-changed Brownian motion, the time change being given by level-crossing times of an independent one-sided stable process with index a. As an application, we prove local asymptotic mixed normality of a statistical model (the critical branching process with immigration) and compare asymptotic behaviour of different estimators for the unknown parameter.