We provide a probabilistic proof of the Stein's factors based on properties of birth and death Markov chains, solving a tantalising puzzle in using Markov chain knowledge to view the celebrated Stein-Chen method for Poisson approximations. This paper complements the work of Barbour (1988) for the case of Poisson random variable approximation. The Stein-Chen method was introduced in Chen (1975) ...

1 Basic aspects of continuous time Markov chains 3 1.1 Markov property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Regular jump chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Holding times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Poisson process . . . . . . . . . . . . . . ....

1. Introduction: In the research area of one dimensional stochastic processes, Markov chains acquire special importance due to large number of applications. One of the simplest possible continuous time Markov chains, namely birth-and-death process arises naturally in many queueing models. Such a Markov chain has an efficient recursive solution for the equilibrium probabilities. Specifically the...

We prove under mild conditions that the Fleming-Viot process selects minimal quasi-stationary distribution for Markov processes with soft killing on non-compact state spaces. Our results are applied to multi-dimensional birth and death processes, continuous time Galton-Watson diffusion killing.

We shall be concerned with the problem of determining the quasistationary distributions of an absorbing continuous-time Markov chain directly from the transition-rate matrix Q. We shall present conditions which ensure that any finite μ-invariant probability measure for Q is a quasistationary distribution. Our results will be illustrated with reference to birth and death processes. QUASISTATIONA...

Reversible jump methods are the most commonly used Markov chain Monte Carlo tool for exploring variable dimension statistical models. Recently, however, an alternative approach based on birth-and-death processes has been proposed by Stephens for mixtures of distributions.We show that the birth-and-death setting can be generalized to include other types of continuous time jumps like split-and-co...

We consider an interacting particle Markov process for Darwinian evolution in an asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate, and a probability μ of mutation at each birth event. We introduce a renormalization parameter K scaling the size of the population, which leads, when K → +∞, to a deterministic dynamics for ...