Absorbing Markov and branching processes with instantaneous resurrection
type: Item
instance of: scholarly article
title: Absorbing Markov and branching processes with instantaneous resurrection
zbMATH Open document ID: 0796.60086
DOI: 10.1016/0304-4149(93)90108-G
author: Anthony G. Pakes
published in: Stochastic Processes and their Applications
publication date: 1994-09-20T00:00:00Z
review text: Modifying the standard Markov branching process so that zero is no longer an absorbing state may be done in several ways. \textit{M. Yamazato} [J. Math. Soc. Japan 27, 479-496 (1975; Zbl 0303.60081)] introduced an exponential holding time at zero followed by a jump to a random state, the process then behaving as before. \textit{A. Chen} and \textit{E. Renshaw} [Probab. Theory Relat. Fields 87, No. 2, 209-240 (1990; Zbl 0695.60080)] studied a process with instantaneous visits to zero and in which the branching part is necessarily explosive; this process is mathematically properly defined but difficult to envisage as a realistic model. Stimulated by this criticism, the present author considers the limit of Yamazato's model as the exponential holding parameter tends to infinity; in this third model the state zero effectively disappears and the modified behaviour now takes place in state 1. The author compares various properties of all three models, such as the recurrence of states. He also extends the notion of instantaneous resurrection from an absorbing barrier to more general Markov processes.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J27
zbMATH DE Number: 488398
zbMATH Keywords: transition functions and generators
zbMATH Keywords: recurrence classification
zbMATH Keywords: Markov branching process
reviewed by: D. R. Grey
MaRDI profile type: Publication
cites work: Continuous-time Markov chains. An applications-oriented approach
cites work: Markov chains with replacement
cites work: Q5674726
cites work: Limit theorems for semi-Markov processes and renewal theory for Markov chains
cites work: Markov branching processes with instantaneous immigration
cites work: Recurrence of Markov branching processes with immigration
cites work: Q5524074
cites work: Markov renewal theory
cites work: Q4109064
cites work: Q5342182
cites work: Q3721531
cites work: Equivalent descriptions of perturbed Markov processes
cites work: Q5727090
cites work: Markov chain models - rarity and exponentiality
cites work: Piecewise Markov Processes
cites work: The age of a Markov process
cites work: Limit theorems for the population size of a birth and death process allowing catastrophes
cites work: Comments on the age distribution of Markov processes
cites work: A note on the classification of Q-processes when Q is not regular
cites work: Perturbation models
cites work: Q5600237
cites work: Some results on continuous time branching processes with state-dependent immigration
description: scientific article; zbMATH DE number 488398