Decay parameter and related properties of \(n\)-type branching processes
type: Item
instance of: scholarly article
title: Decay parameter and related properties of \(n\)-type branching processes
zbMATH Open document ID: 1264.60065
author: Juan Wang
author: Junping Li
published in: Science China. Mathematics
publication date: 2013-01-28T00:00:00Z
review text: The authors consider in this paper decay properties including the decay parameter, invariant measures, invariant vectors and quasistationary distributions for \(n\)-type Markov branching processes (\(n\)TMBPs) on the basis of the 1-type Markov branching processes and 2-type Markov branching processes. After the Introduction (Section 1), in Section 2, some deep properties of the generating functions for \(n\)-type branching (\(n\)TB) \(q\)-matrix are investigated. The properties of the generating functions \(\{B_i(u_1,\dotsc, u_n)\}\), \(i= 1,\dotsc, n\), defined at the beginning of the section are discussed. Seven lemmas are proved in this sense. In Section 3, the authors study the decay parameter for \(n\)TMBPs. Thus it is proved that, for any given \(n\)TB \(q\)-matrix \(Q\) defined in Definition 1.1, there always exists exactly one \(Q\)-process satisfying the Kolmogorov forward equation. In this sense, Theorem 3.1 proves the uniqueness property of \(n\)TMBPs; and Theorem 3.2 gives a lower bound of the decay parameter. Then the exact value of the decay parameter is presented in Theorem 3.3. In the last section, the \(\lambda_C\)-invariant vectors/measures and quasistationary distribution of \(n\)TBs are considered. In this way, Theorem 4.1 gives a \(\lambda_C\)-invariant vector for \(Q\) (or for \(P(t)\)) on \(C\); and, in Theorem 4.2, the \(\lambda_C\)-invariant measures for \(Q\) (or for \(P(t)\)) on \(C\) are considered. Then, Theorem 4.3 contains the quasistationary distributions for \(P(t)\) on \(C\). The paper is very good and instructive.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 6131920
zbMATH Keywords: \(n\)-type Markov branching process
zbMATH Keywords: decay parameter
zbMATH Keywords: invariant measures
zbMATH Keywords: invariant vectors
zbMATH Keywords: quasistationary distributions
reviewed by: Gabriel V. Orman
MaRDI profile type: Publication
full work available at URL: s11425-012-4466-z
OpenAlex ID: W2072374169
cites work: Continuous-time Markov chains. An applications-oriented approach
cites work: Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process
cites work: Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes
cites work: On quasi-stationary distributions in absorbing continuous-time finite Markov chains
cites work: Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains
cites work: Q5727090
cites work: Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous time
cites work: The Exponential Decay of Markov Transition Probabilities
cites work: Decay parameter and related properties of 2-type branching processes
cites work: On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains
cites work: On the equivalence of \(\mu\)-invariant measures for the minimal process and its q-matrix
cites work: Reversibility, invariance and μ-invariance
cites work: The determination of quasistationary distributions directly from the transition rates of an absorbing Markov chain
cites work: Quasistationary distributions for continuous time Markov chains when absorption is not certain
cites work: SOME ERGODIC PROPERTIES OF THE FELLER MINIMAL PROCESS
cites work: GEOMETRIC ERGODICITY IN DENUMERABLE MARKOV CHAINS
DOI: 10.1007/S11425-012-4466-Z
description: scientific article; zbMATH DE number 6131920