TY - JOUR

T1 - Growth of integral transforms and extinction in critical Galton-Watson processes

AU - Tokarev, Daniel

PY - 2008

Y1 - 2008

N2 - The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n

AB - The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n

UR - http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.jap/1214950361&page=record

U2 - 10.1239/jap/1214950361

DO - 10.1239/jap/1214950361

M3 - Article

VL - 45

SP - 472

EP - 480

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 2

ER -