Limiting Distributions For Critical Multitype Branching Processes

Limiting Distributions for Critical Multitype Branching Processes

Limiting Distributions for Critical Multitype Branching Processes with Discrete Time

The theory of branching processes treats the multiplication of objects subject to laws of chance. It has been applied to such phenomena as bacterial fission, cosmic ray showers, and neutron chain reactions. This report describes the probability law governing the descendants of a single object after many generations, if on the average an object has just one surviving child (in the language of reactors, this is the critical case). These results could be useful in the study of controlled nuclear reactions. (Author).

Branching Processes

The purpose of this book is to give a unified treatment of the limit theory of branching processes. Since the publication of the important book of T E. Harris (Theory of Branching Processes, Springer, 1963) the subject has developed and matured significantly. Many of the classical limit laws are now known in their sharpest form, and there are new proofs that give insight into the results. Our work deals primarily with this decade, and thus has very little overlap with that of Harris. Only enough material is repeated to make the treatment essentially self-contained. For example, certain foundational questions on the construction of processes, to which we have nothing new to add, are not developed. There is a natural classification of branching processes according to their criticality condition, their time parameter, the single or multi-type particle cases, the Markovian or non-Markovian character of the pro cess, etc. We have tried to avoid the rather uneconomical and un enlightening approach of treating these categories independently, and by a series of similar but increasingly complicated techniques. The basic Galton-Watson process is developed in great detail in Chapters I and II.