Intertwining of birth-and-death processes

TL;DR

This paper explores the relationship between birth-and-death processes and their eigenvalues, extending probabilistic coupling methods to include a new process that always lags behind the original, providing deeper insight into their intertwining properties.

Contribution

It introduces a novel coupling construction involving a third process with ordered eigenvalue-based birth rates that always lags behind the original process.

Findings

Constructed a process that always lags behind the original process.

Extended probabilistic coupling methods to include a third process.

Provided new insights into the intertwining of birth-and-death processes.

Abstract

It has been known for a long time that for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues, ordered from high to low, and whose death rates are zero, in such a way that the latter process is always ahead of the former, and both arrive at the same time at the given level. In this note, we extend their methods by constructing a third process, whose birth rates are the negatives of the eigenvalues ordered from low to high and whose death rates are zero, which always lags behind the original process and also arrives at the same time.