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Limit theorems for the population size of a birth and death process allowing catastrophes. (English) Zbl 0642.92012

A quite general linear birth and death process with catastrophes X t (BDCP), which is a continuous-time right-continuous random walk on { 0,1,2,...} with instantaneous jump rates proportional to X t , is considered. I. I. Ezhov and V. N. Reshetnyak [Ukr. Mat. Zh. 35, No.1, 31-36 (1983; Zbl 0531.60081); English translation in Ukr. Math. J. 35, 28-33 (1983)] formulated that process and studied the tail behaviour of the extinction time T, obtaining, in the way, an identity relating P(T>t|X 0 =1) to a similar quantity for a certain Markov branching process (MBP) X ^ t . The connection is shown to be deeper, the transition probabilities of the two processes also being closely related.

Such a relation is used to obtain new properties of the BDCP or similar properties to those obtained by Ezhov and Reshetnyak under weaker conditions. In particular, E(X t |X 0 =i), P(X t >0|X 0 =i) are studied, along with their asymptotic behaviour as t. The limiting distribution of X t is obtained, in some cases conditioned on non-extinction. A Q-process corresponding to the BDCP is shown to exist under mild regularity assumptions and its limiting behaviour is studied.

Reviewer: C.A.Braumann

MSC:
92D25Population dynamics (general)
60J27Continuous-time Markov processes on discrete state spaces
60J80Branching processes
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