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The supercritical birth, death and catastrophe process: Limit theorems on the set of extinction. (English) Zbl 0726.60084
Summary: The stationary conditional quasi-stationary distribution of the linear birth, death and catastrophe process is shown to exist iff the decrement distribution has a finite second order moment. Conditional limit theorems for the population size are found when this moment is infinite and a regular variation condition is satisfied. The relevance of the results in this paper to the general theory of quasi-stationary distributions is discussed.

MSC:
60J80Branching processes
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References:
[1] Asmussen, S.; Hering, H.: Branching processes. (1983) · Zbl 0516.60095
[2] Athreya, K. B.; Ney, P. E.: Branching processes. (1972) · Zbl 0259.60002
[3] Barbour, A. D.; Pakes, A. G.: Limit theorems for the simple branching process allowing immigration, II. The case of infinite offspring mean. Adv. appl. Prob. 11, 63-72 (1979) · Zbl 0401.60078
[4] Doney, R. A.: A note on a condition satisfied by certain random walks. J. appl. Prob. 14, 843-849 (1977) · Zbl 0378.60057
[5] Erickson, K. B.: Strong renewal theorems with infinite mean. Trans. amer. Math. soc. 151, 263-291 (1970) · Zbl 0212.51601
[6] Feller, W.: 2nd edition an introduction to probability theory and its applications. An introduction to probability theory and its applications (1971) · Zbl 0219.60003
[7] Flaspohler, D. C.: Quasi-stationary distibutions for absorbing continuous-time denumerable Markov chains. Ann. inst. Statist. math. 26, 351-356 (1974) · Zbl 0344.60039
[8] Pakes, A. G.: The Markov branching-catastrophe process. Stochastic processes appl. 23, 1-33 (1986) · Zbl 0633.92014
[9] Pakes, A. G.: Limit theorems for the population size of a birth and death process allowing catastrophes. J. math. Biology 25, 307-325 (1987) · Zbl 0642.92012
[10] Pakes, A. G.: The supercritical birth, death and catastrophe process: limit theorems on the set of non-extinction. J. math. Biology 26, 405-420 (1988) · Zbl 0721.60093
[11] Pakes, A. G.: Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements. Adv. appl. Prob. 21 (1989) · Zbl 0671.60078
[12] Pollett, P. K.: On the equivalence of ${\mu}$-invariant measures for the minimal process and its q-matrix. Stochastic process. Appl. 22, 203-221 (1986) · Zbl 0658.60099
[13] Pollett, P. K.: Reversibility, invariance and ${\mu}$-invariance. Adv .Appl. Prob. 20, 600-621 (1988) · Zbl 0654.60058
[14] Seneta, E.: Regularly varying functions. Lecture notes in mathematics 508 (1976) · Zbl 0324.26002
[15] Vere-Jones, D.: Some limit theorems for evanescent processes. Aust. J. Statist. 2, 67-78 (1969) · Zbl 0188.23302