{
  "id": "2409.05018",
  "version": "v1",
  "published": "2024-09-08T08:16:27.000Z",
  "updated": "2024-09-08T08:16:27.000Z",
  "title": "Approximation of birth-death processes",
  "authors": [
    "Liping Li"
  ],
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-08T08:16:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuous-time markov chain",
      "approximation methods",
      "convergence",
      "one-point compactification space",
      "strong markov property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}