Letters to the Editor on the Asymptotic Behaviour of the Extinction Time of the Simple Branching Process

The time to extinction of a subcritical Galton-Watson branching process and the time of last mutation of its infinite-alleles version are maxima of independent random variables having an upper tail of geometric type, and hence they are not attracted to any extreme value distribution. It is shown that Anderson's asymptotic results for maxima of discrete variates are applicable, and this rectifies a false assertion made in respect to the infinite-alleles simple branching process. MAXIMA; EXTREME VALUE DISTRIBUTION Let L be the time of last mutation in the infinite-alleles version of the simple branching process as described by Griffiths and Pakes (1988). The underlying branching process is denoted by {Zn:n ~O} andf(.) is its offspring probability generating function (p.g.f.), {3(.) is the p.g.f. of the total number of births and u is the probability that a newly born individual mutates to a new allele. It was shown in Lemma 3.1 of Griffiths and Pakes (1988) that the distribution function of L is given by P(L ~ n I z, = i) = [fn(c5)l, where c5 = {3(1 u) and fn(.) is the n-fold functional interate of f(.). Hence L has the same distribution as the maximum of i independent copies of L evaluated for a family tree having a single ancestor. If c5 = 0 we recover the distribution function of the time T to extinction of {Zn}. We shall be concerned with the subcritical case m = Et(Zt) < 1. In this case the conditional distribution of Zn, given T > n, has a weak limit whose p.g.f. is denoted by Q(.); see Athreya and Ney (1972). Let ~=logm-t and 1jJ(.) be the inverse of Q(.). Griffiths and Pakes (1988) purported to prove (theorem 3.1(ii)) that lim p;(i1jJ(exp-~L) ~x) = 1exp [-x(lQ(c5))]. ;-+00 This is not true (mea culpa). When A= E(Ztlog+Zt) < 00 then c = Q'(l-) < 00 and then the asserted result could be rewritten to say that a centring sequence b(i) exists such that {L b(i)} converges in law to the Gumbel extreme value distribution. However, since (1) Pt(L > n)1 Pt(L > n + 1) = (1 fn(c5))/(1 fn+t( c5))~ 11m, the well-known necessary condition for attraction to the Gumbel distribution is violated; see Received 8 December 1988. * Postal address: Department of Mathematics, The University of Western Australia, Nedlands, WA6009, Australia. 470 terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0001867800018644 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 20 Aug 2017 at 03:05:32, subject to the Cambridge Core Letters to the editor 471 Resnick (1987), p. 45. The error in the Griffiths and Pakes proof occurs where they let i and n be such that i(1fn(O))~x; this is not possible for all x. The situation can be partially salvaged because (1) is necessary and sufficient for Anderson's (1970) Theorem 2 to be applicable. Recall that c > 1 and c < 00 iff A< 00. Theorem 1. If m < 1 there is an increasing sequence of constants {b(i)} satisfying (2) b(i) C-Ilog i (i~ (0) and exp (_e-{;(X-I») ~ lim inf P;(L b(i) ~x) ~ lim sup P;(L b(i) ~x) ~ exp (-e-~). ;-+00 ;-+00