Limiting Distributions for Multitype Branching Processes

• Published in: Stochastic analysis and applications Vol. 28; no. 6; pp. 1040 - 1060
• Main Authors: Yakovlev, Andrei Y., Yanev, Nikolay M.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 01.01.2010; Taylor & Francis
• Subjects: Asymptotic properties; Biology; Constraining; Counting; Exact sciences and technology; Infinity; Limit theorems; Markov processes; Mathematics; Multitype branching processes; Normality; Primary 60J80, 60J85; Probability and statistics; Probability theory and stochastic processes; Relative frequencies; Sciences and techniques of general use; Secondary 62P10, 92D25; Stochastic analysis; Stochastic processes; Stochasticity; Markov process; Stochastic analysis; Branching process; Limit distribution; Asymptotic normality; Asymptotic behavior; Law of large numbers; Biology; Discrete time processes; Multivariate analysis; multivariate asymptotic normality; relative frequencies; limit theorems; multitype branching processes
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1080/07362994.2010.515486

In this article, the asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity. Some limiting distributions are obtained as well as multivariate asymptotic normality is proved. The article also considers the relative frequencies of distinct types of individuals motivated by applications in the field of cell biology. We obtained non-random limits for the frequencies and multivariate asymptotic normality when the initial number of ancestors is large and the time of observation increases to infinity. In fact this paper continues the investigations of Yakovlev and Yanev [ 32 ] where the time was fixed. The new obtained limiting results are of special interest for cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement.