Analysis of a Recurrence Related to Critical Nonhomogeneous Branching Processes

Abstract

Some classes of controlled branching processes (with nonhomo-geneous migration or with nonhomo-geneous state-dependent immigration) lead in the critical case to a recurrence for the extinction probabilities. Under some additional conditions it is known that this recurrence depends on some parameter β and converges for 0 < β < 1. Now we show that the recurrence does converge for all positive values of the parameter β, which leads to an extension of some limit theorems for the corresponding branching processes. We also give a generalization of the recurrence and an asymptotic analysis of its behavior.

Keywords:

• Branching processes
• Extinction probabilty
• Random control function

Mathematics Subject Classification:

• 60J80

ACKNOWLEDGMENT

The authors wish to thank F. T. Bruss for useful comments on the first version of this paper. The pertinent remarks of the referee led to corrections and improvement in the presentation.

Notes

¹Here we use the indicator function notation proposed by Knuth et al. [23].