Final probabilities for modified branching processes
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B.A. Sevastyanov
Abstract
Any modified branching process ℬ* is constructed by means of two Galton-Watson processes ℬ0, ℬ1, and a fixed finite set S of positive integers. The number of particles μ*(t) of the process ℬ* at time instants t = 0, 1, 2, ... evolves as follows. If μ*(t) ∈ S, then each of the μ*(t) particles independently of each other produces an offspring according to the law of the branching process ℬ1, and if μ*(t) ∉ S, then the birth of particles obeys the law of the process ℬ0. Along with active, breeding particles, in the processes ℬ0 and ℬ1 a random amount of final particles emerges, which do not participate in the process evolution but accumulate and constitute some final amount ηn after the process extinction, where n is the initial number of active particles. It is known that in a critical branching process, under some conditions, the distribution of the random variable ηn/n2 as n → ∞ converges to the stable distribution law with parameter α = 1/2. In this paper, we demonstrate that this property of the distribution of final particles remains true for the modified branching process ℬ*. We also show that in this limit theorem the number of final particles can be replaced by a certain final non-negative random variable TJn that characterises the final state of the branching process.
© 2016 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Editorial
- Final probabilities for modified branching processes
- Non-interference models and subliminal channels
- Monic polylinear shift registers and their periods
- Limit distributions of the maximum size of a tree in a random recursive forest
- On the asymptotic behaviour of probabilities of large deviations for the negative polynomial distribution
- On the functional complexity of a two-dimensional interval search problem
- On analytical conjugacy of Ton chard polynomials and the polynomials quasi-orthogonal to them
- Forthcoming Papers
- Contents
Articles in the same Issue
- Editorial
- Final probabilities for modified branching processes
- Non-interference models and subliminal channels
- Monic polylinear shift registers and their periods
- Limit distributions of the maximum size of a tree in a random recursive forest
- On the asymptotic behaviour of probabilities of large deviations for the negative polynomial distribution
- On the functional complexity of a two-dimensional interval search problem
- On analytical conjugacy of Ton chard polynomials and the polynomials quasi-orthogonal to them
- Forthcoming Papers
- Contents
