Limiting Distributions for Multitype Branching Processes

Abstract

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.

Keywords:

• Limit theorems
• Multitype branching processes
• Relative frequencies

Mathematics Subject Classification:

• Primary 60J80, 60J85
• Secondary 62P10, 92D25

This article is supported by NIH/NINDS grant NS39511, NIH/NCI R01 grant CA134839, and NIH grant N01-AI-050020.

This article was prepared while N. Yanev was a visiting professor in the Department of Biostatistics and Computational Biology, University of Rochester, and he is grateful for hospitality and inspiring communications with his colleagues.

Notes

communications with his colleagues.

Additional information

Notes on contributors

Andrei Y. Yakovlev

Professor Andrei Yakovlev died suddenly on February 27, 2008, when our work was in preparation. This article is a tribute to his stimulating ideas and the friendship that we shared in our collaboration.