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On survival probabilities and extinction times for some branching processes

1971; Cambridge University Press; Volume: 8; Issue: 4 Linguagem: Inglês

10.2307/3212229

1475-6072

Edward Pollak,

Evolution and Genetic Dynamics

Summary We consider branching processes for which the first three moments of the distribution of offspring exist. Let f ( t ) and z be, respectively, the generating function of the distribution of offspring and the smallest positive root of the equation f ( t ) = t. Then if M = f' ( z ) and f n ( t ) is equal to the generating function of the distribution of n th generation descendants of a single individual, it is known that, quite generally, ( z – f n (0)) / M n tends toward a constant as n increases. A method is derived for obtaining upper and lower bounds for this constant, which gives an exact solution when there is a geometric distribution of offspring and good bounds when there are Poisson or negative binomial offspring distributions. With some further calculations, one can also obtain finite upper and lower bounds for the mean time to extinction of a line descended from an individual, given there is extinction. These bound apply even if z is less than 1. Numerical values are given for the Poisson and negative binomial cases.