出生-死亡过程的近似值

Liping Li
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摘要

出生-死亡过程是一种特殊的连续时间马尔可夫链,其索引集为 $\mathbb{N}$。它的解析矩阵可以用一组参数$(\gamma, \beta, \nu)$来完全描述,其中$\gamma$和$\beta$是非负常量,$\nu$是$\mathbb{N}$上的正量度。通过使用雷-奈特致密化,出生-死亡过程可以被看作是在一点致密化空间 $\overline\{mathbb{N}}_{\partial}$ 上具有强马尔可夫性质的 c\`adl\`ag 过程,其中包括一个额外的墓地点 $\partial$ 。从某种意义上说,决定出生-死亡过程的三个参数分别对应于它在用于一点紧凑化的(one-point compactification)$\infty$处的杀戮、反射和跳跃行为。一般来说,要清晰地描述出生-死亡过程的轨迹,尤其是在$|\nu|=\infty$的病理情况下,是很有挑战性的。本文旨在通过使用近似方法研究出生-死亡过程来解决这一问题。具体来说,我们将用更容易理解的更简单的出生-死亡过程来近似出生-死亡过程。对于两种典型的近似方法,我们的主要结果证明了由近似过程引起的概率度量序列在所有 c\`adl\`ag 函数空间上的弱收敛性。这种收敛性明显强于连续时间马尔可夫链理论中通常考虑的过渡矩阵的收敛性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Approximation of birth-death processes
The birth-death process is a special type of continuous-time Markov chain with index set $\mathbb{N}$. Its resolvent matrix can be fully characterized by a set of parameters $(\gamma, \beta, \nu)$, where $\gamma$ and $\beta$ are non-negative constants, and $\nu$ is a positive measure on $\mathbb{N}$. By employing the Ray-Knight compactification, the birth-death process can be realized as a c\`adl\`ag process with strong Markov property on the one-point compactification space $\overline{\mathbb{N}}_{\partial}$, which includes an additional cemetery point $\partial$. In a certain sense, the three parameters that determine the birth-death process correspond to its killing, reflecting, and jumping behaviors at $\infty$ used for the one-point compactification, respectively. In general, providing a clear description of the trajectories of a birth-death process, especially in the pathological case where $|\nu|=\infty$, is challenging. This paper aims to address this issue by studying the birth-death process using approximation methods. Specifically, we will approximate the birth-death process with simpler birth-death processes that are easier to comprehend. For two typical approximation methods, our main results establish the weak convergence of a sequence of probability measures, which are induced by the approximating processes, on the space of all c\`adl\`ag functions. This type of convergence is significantly stronger than the convergence of transition matrices typically considered in the theory of continuous-time Markov chains.
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