{"title":"Approximation of birth-death processes","authors":"Liping Li","doi":"arxiv-2409.05018","DOIUrl":null,"url":null,"abstract":"The birth-death process is a special type of continuous-time Markov chain\nwith index set $\\mathbb{N}$. Its resolvent matrix can be fully characterized by\na set of parameters $(\\gamma, \\beta, \\nu)$, where $\\gamma$ and $\\beta$ are\nnon-negative constants, and $\\nu$ is a positive measure on $\\mathbb{N}$. By\nemploying the Ray-Knight compactification, the birth-death process can be\nrealized as a c\\`adl\\`ag process with strong Markov property on the one-point\ncompactification space $\\overline{\\mathbb{N}}_{\\partial}$, which includes an\nadditional cemetery point $\\partial$. In a certain sense, the three parameters\nthat determine the birth-death process correspond to its killing, reflecting,\nand jumping behaviors at $\\infty$ used for the one-point compactification,\nrespectively. In general, providing a clear description of the trajectories of a\nbirth-death process, especially in the pathological case where $|\\nu|=\\infty$,\nis challenging. This paper aims to address this issue by studying the\nbirth-death process using approximation methods. Specifically, we will\napproximate the birth-death process with simpler birth-death processes that are\neasier to comprehend. For two typical approximation methods, our main results\nestablish the weak convergence of a sequence of probability measures, which are\ninduced by the approximating processes, on the space of all c\\`adl\\`ag\nfunctions. This type of convergence is significantly stronger than the\nconvergence of transition matrices typically considered in the theory of\ncontinuous-time Markov chains.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05018","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.