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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 $X\sb t(BDCP)$, which is a continuous-time right-continuous random walk on $\{$ 0,1,2,...$\}$ with instantaneous jump rates proportional to $X\sb t$, is considered. {\it I. I. Ezhov} and {\it 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 $P(T>t\vert X\sb 0=1)$ to a similar quantity for a certain Markov branching process (MBP) $\hat X\sb t$. 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, $E(X\sb t\vert X\sb 0=i)$, $P(X\sb t>0\vert X\sb 0=i)$ are studied, along with their asymptotic behaviour as $t\to \infty$. The limiting distribution of $X\sb t$ 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

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