Probabilistic approach in weighted Markov branching processes

• Published in: Statistics & probability letters Vol. 78; no. 6; pp. 771 - 779
• Main Authors: Chen, Anyue, Li, Junping, Ramesh, N.I.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.04.2008; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Compound Poisson process; Compound Poisson process Hitting times Markov branching process Random time change Uniqueness Weighted Markov branching process; Exact sciences and technology; General topics; Hitting times; Inference from stochastic processes; time series analysis; Markov branching process; Markov processes; Mathematics; Probability and statistics; Probability theory and stochastic processes; Random time change; Sciences and techniques of general use; Statistics; Uniqueness; Weighted Markov branching process; Weighted Markov branching process; Hitting times; Uniqueness; Compound Poisson process; Markov branching process; Random time change; secondary 60 J80; primary 60 J27; Markov process; Compound process; Compound Poisson process; Hitting times; Markov branching process; Random time change; Uniqueness; Weighted Markov branching process; Conditional distribution; Probabilistic approach; Extinction; Probability theory; Explosions; primary 60 J27; secondary 60 J80; Stochastic process; Poisson process; Statistical method; Branching process; Survival time
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2007.09.043

This note provides a new probabilistic approach in discussing the weighted Markov branching process (WMBP) which is a natural generalisation of the ordinary Markov branching process. Using this approach, some important characteristics regarding the hitting times of such processes can be easily obtained. In particular, the closed forms for the mean extinction time and conditional mean extinction time are presented. The explosion behaviour of the process is investigated and the mean explosion time is derived. The mean global holding time and the mean total survival time are also obtained. The close link between these newly developed processes and the well-known compound Poisson processes is investigated. It is revealed that any weighted Markov branching process (WMBP) is a random time change of a compound Poisson process.