Mathematics > Probability
[Submitted on 22 Nov 2022 (v1), last revised 18 Aug 2024 (this version, v3)]
Title:Ray-Knight compactification of birth and death processes
View PDF HTML (experimental)Abstract:A birth and death process is a continuous-time Markov chain with the minimal state space $\mathbb N$, whose transition matrix is standard and whose density matrix is the given birth-death matrix. Birth and death process is unique if and only if $\infty$ is an entrance or natural. When $\infty$ is neither an entrance nor natural, there are two ways in the literature to obtain all birth and death processes. The first one is an analytic treatment proposed by Feller in 1959, and the second one is a probabilistic construction completed by Wang in 1958.
In this paper we will give another way to study birth and death processes using the Ray-Knight compactification. This way has the advantage of both the analytic and probabilistic treatments above. By applying the Ray-Knight compactification, every birth and death process can be modified into a càdlàg Ray process on $\mathbb N\cup \{\infty\}\cup\{\partial\}$, which is either a Doob processes or a Feller $Q$-process. Every birth and death process in the second class has a modification that is a Feller process on $\mathbb N\cup\{\infty\}\cup \{\partial\}$. We will derive the expression of its infinitesimal generator, which explains its boundary behaviours at $\infty$. Furthermore, by utilizing transformations of killing and Ikeda-Nagasawa-Watanabe's piecing out, we will also provide a probabilistic construction for birth and death processes. This construction relies on a triple determining the resolvent matrix introduced by Wang and Yang in their work (1992')}.
Submission history
From: Liping Li [view email][v1] Tue, 22 Nov 2022 16:04:23 UTC (35 KB)
[v2] Wed, 1 Nov 2023 01:11:16 UTC (43 KB)
[v3] Sun, 18 Aug 2024 05:24:06 UTC (44 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.