Relative frequencies in multitype branching processes

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.Comment: Published in at http://dx.doi.org/10.1214/08-AAP539 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Complete Classification of Catalytic Branching Processes

“…So, assume that ρ(D) > 1. Applying Theorem 2.1, item (iii), in [12] to the system of renewal equations (32) we get…”

“…Assume that ρ(D) = 1 or, equivalently, that the unique solution to ρ(D(ν)) = 1 is ν = 0. Applying Theorem 2.1, item (iii), in [12] to (32) we come to the following relation…”

“…Mention in passing some recent works on multitype Bellman-Harris processes, see, e.g., [18], [30] and [32].…”

Полная Классификация Каталитических Ветвящихся Процессов

“…Укажем лишь несколько недавних работ, относящихся к многотипным процес-сам Беллмана-Харриса, например, [5], [25] и [33].…”

“…Если марковская цепь η возвратна, то соответствующее соотношение следует из (35) и теоремы 2.2, пункт (ii), из [28]. Когда n > 1, эта теорема также приводит к (30) ввиду системы уравнений (33). Заметим, что в силу (39) функция C 1 (w i ) (а следовательно, и C n (w i ) при n > 1) строго положительна для каждого i = 1, .…”