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A note on the asymptotic behaviour of the extinction probability in supercritical population-size-dependent branching processes with independent and identically distributed random environments

Published online by Cambridge University Press:  14 July 2016

Lu Zhunwei*
Affiliation:
Taiyuan University of Technology
Peter Jagers*
Affiliation:
Chalmers University of Technology and Göteborg University
*
Postal address: Department of Mathematics, Taiyuan University of Technology, 030024, Taiyuan, Shanxi Province, P. R. China. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology and Göteborg University, SE-412 96, Göteborg, Sweden. Email address: [email protected]

Abstract

In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 < α 1α 0 < + ∞ and 0 < C 1, C 2 < + ∞ such that the extinction probability starting with k individuals is bounded below by C 1 k -α 0 and above by C 2 k -α 1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2004 

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