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. 1. Introduction. The paper deals with multitype branching processes with discrete or continuous time assuming only the usual independence of the individual evolutions. Such processes are considered from a new perspective: modeling of the relative frequencies (proportions, fractions) of different types of individuals (instead of their absolute numbers) as functions of time. Notwithstanding other possible applications, we will use the term "cell" instead of referring to an abstract individual or particle throughout this paper to emphasize its special relevance to cell biology. It is well known (see, e.g., Athreya and Ney [1] and Mode [19] ) that in the positively regular and supercritical case the multitype population vector
Relative frequencies of types were studied in REF .
Relative frequencies in multitype branching processes
Annals of Applied Probability 2009, Vol. 19, No. 1, 1-14
Mathematics Biology