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Degeneration, probability of

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The probability of no particles being left in a branching process at an epoch . Let be the number of particles in a branching process with one type of particles. The probability of degeneration

does not decrease as increases; the value

is called the probability of degeneration in infinite time or simply the probability of degeneration. If is the time elapsing from the beginning of the process to the epoch of disappearance of the last particle, then and . The rate of convergence of to as has been studied for various models of branching processes.


Comments

The probability of degeneration is more commonly called the probability of extinction (in infinite time).

References

[a1] P.E. Ney, K.B. Athreya, "Branching processes" , Springer (1972)
[a2] T.E. Harris, "The theory of branching processes" , Springer (1963)
How to Cite This Entry:
Degeneration, probability of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degeneration,_probability_of&oldid=46617
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article