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
This paper demonstrates a new regeneration processes technology making use of positive stable distributions. We study the asymptotic behavior of branching processes with a randomly controlled migration component. Using the new method, we confirm some known results and establish new limit theorems that hold in a more general setting.
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