Relative frequencies in multitype branching processes

TL;DR

This paper proves that the relative frequencies of different types in multitype branching processes tend to a multivariate normal distribution as the initial population grows large, aiding statistical inference in biological studies.

Contribution

It establishes the asymptotic normality of relative frequencies in multitype branching processes under broad conditions, extending previous results and enabling new statistical applications.

Findings

Relative frequencies are asymptotically multivariate normal.

Result holds for any finite-type branching process with independent evolutions.

Facilitates statistical inference in cell biology applications.

Abstract

This paper considers the relative frequencies of distinct types of individuals in multitype branching processes. We prove that the frequencies are asymptotically multivariate normal when the initial number of ancestors is large and the time of observation is fixed. The result is valid for any branching process with a finite number of types; the only assumption required is that of independent individual evolutions. The problem under consideration is motivated by applications in the area of cell biology. Specifically, the reported limiting results are of advantage in cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement. Relevant statistical applications are discussed in the context of asymptotic maximum likelihood inference for multitype branching processes.