UND INFORMATIK. Asexual Versus Promiscuous Bisexual Galton-Watson Processes: The Extinction Probability Ratio. G. Alsmeyer und U.

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1 ANGEWANDTE MATHEMATIK UND INFORMATIK Asexual Versus Promiscuous Bisexual Galton-Watson Processes: The Extinction Probability Ratio G. Alsmeyer und U. Rösler Universität Münster und Universität Kiel Bericht Nr. 27/00-S UNIVERSITÄT MÜNSTER

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3 1 Asexual Versus Promiscuous Bisexual Galton-Watson Processes: The Extinction Probability Ratio Gerold Alsmeyer Institut für Mathematische Statistik Fachbereich Mathematik Westfälische Wilhelms-Universität Münster Einsteinstraße 62 D Münster, Germany Uwe Rösler Mathematisches Seminar Christian-Albrechts-Universität Kiel Ludewig-Meyn-Straße 4 D Kiel, Germany We consider the supercritical bisexual Galton-Watson process BGWP with promiscuous mating, that is a branching process which behaves like an ordinary supercritical Galton-Watson process GWP as long as at least one male is born in each generation. For a certain example, it was pointed out by Daley et al. [7] that the extinction probability of such a BGWP apparently behaves like a constant times the respective probability of its asexual counterpart where males do not matter if the number of ancestors grows to infinity. In an earlier paper [1] we provided general upper and lower bounds for the ratio between both extinction probabilities and also numerical results that seemed to confirm the convergence of that ratio. However, theoretical considerations rather led us to the conjecture that this does not generally hold. The present article turns this conjecture into a rigorous result. The key step in our analysis is to identify the extinction probability ratio as a certain functional of a subcritical ordinary GWP and to prove its continuity as a function of the number of ancestors in a suitable topology associated with the Martin entrance boundary of that GWP. AMS 2000 subject classifications. 60J80, 60J45, 60J50. Keywords and phrases. Bisexual Galton-Watson process, promiscuous mating, extinction probability, quasi invariant measure, time reversal.

4 2 1. Introduction and main results The bisexual Galton-Watson process with promiscuous mating Z n, shortly called promiscuous BGWP, is defined as follows: Consider a two sex population process Zn F, Zn M whose n-th generation consists of Zn F females and Zn M males. Females within one generation reproduce according to an ordinary two-type Galton-Watson process GWP with product reproduction law p F p M as long as at least one male is alive. Plainly, p F =p F j j 0 and p M =p M j j 0 describe the number of female, respectively male offspring. We are therefore given def = Zn F 1 0, Zn M Z n mating units in the n-th generation, the pertinent mating function being ζx, =x1 0,. The formal definition of Zn F, Zn M thus takes the form Z F n+1, Z M n+1 = Z n j=1 ξ n,j,η n,j 1.1 with i.i.d. random vectors ξ n,j,η n,j,,j 1 with common distribution p F p M. Bisexual GWPs with various mating functions were introduced by Daley [6] and further investigated in a series of papers [5],[7],[8],[9]. The present article is a continuation of [1] where we compared in some detail the extinctive behavior of a promiscuous BGWP Z n with that of its asexual counterpart, henceforth denoted by F n. Let P j be such that P j Z 0 = F 0 = j = 1 for each j 1 and define the extinction probability function qj def = P j Z n = 0 for some n 0, j IN 0, pertaining to Z n. Plainly, the reproduction law of the ordinary GWP F n is p F, its extinction probability function q j for some q [0, 1]. We are interested in the supercritical case when qj < 1 for all j 1, a standing assumption hereafter. For the promiscuous BGWP this is easily seen to be equivalent to p M 0 < 1 and µ def = j 1 jpf j > 1. Hence F n is also supercritical and its extinction probability q less than 1. A numerical study in [7] showed for the case where p F and p M are Poisson with mean 1.2 that the extinction probability ratio rk def = qk q k, k IN, apparently converges very rapidly to approximately On the other hand, they had no theoretical justification for this phenomenon and our analysis in [1] indeed showed that this can neither be given shortly nor by easy arguments. Let ˆP k = P k F n 0 with expectation operator Êk and put κ def = p M 0. By exploiting a functional equation for rk, namely rk = k κ + 1 κ k ÊkrF q

5 3 for each k 0, we were led in [1] to lower and upper bounds for rk depending on the model parameters. Numerical studies for various sets of parameters further confirmed the observation of Daley et al. that rk rapidly stabilizes for increasing k if κ<q. However, based upon arguments beyond the scope of that article, we conjectured that rk may actually not always converge but oscillate very slowly, a near-constancy phenomenon also encountered for the so-called Harris function of certain supercritical ordinary GWP, see e.g. [4]. The main result of the present article, Theorem 2.1, shows that this conjecture is correct. The proof is based on potential theory for subcritical GWPs which is therefore shortly reviewed from [3] in the Section 3. Iterating equation 1.2 leads to the fundamental identity see 3.12 in [1] rk = k κ + q Êk τ j=1 κ q Fj j 1 1 κ F i 1.3 where τ denotes the extinction time of F n. Note that, under ˆP k, F n forms an ordinary subcritical GWP with k ancestors, reproduction mean ˆµ = f q, offspring distribution ˆp F =q j 1 p F j j 0and offspring generating function ˆfs =q 1 fsq, f the generating function of p F, see [3, p. 37f]. Note that ˆP 1 F 1 > 1 = q j 1 p F j > j 2 and that q<1 clearly implies the X log X-condition for F n under the ˆP k, i.e. Ê k F 1 log F 1 <. 1.5 Our main concern hereafter will be the case 0 <κ<qwhere the near-constancy phenomenon turns up, but we will also provide a result for the case κ = q Theorem 2.2 below. If κ>q, we already gave a satisfactory answer in [1], Corollary 3.2 which states that κ k qk converges to 1 at an exponential rate. 2. Main results A look at identity 1.3 shows that its further investigation does no longer require to deal with the original model of a promiscuous BGWP from which it came out. We may rather adopt the viewpoint of dealing with a certain functional in two arguments, κ and q, ofan ordinary subcritical GWP. We will therefore simplify our previous notation and use the one for Galton-Watson branching processes in [3] to which we will frequently refer. So from now on let Z n be a subcritical GWP with offspring distribution p j j 0, offspring generating function fs = j 0 p js j, reproduction mean µ = f 1 < 1 and extinction time τ. Notice that now fq q. For each k 1, P k shall denote the probability measure under which

6 4 Z 0 = k. Ifk= 1, we also write P instead of P 1. We further assume see also 1.4 and 1.5 p 1 > 0, p 0 +p 1 < and the X log X-condition EZ 1 log Z 1 <. 2.2 These conditions will in fact be needed in the course of our subsequent analysis. The first condition together with p 0 > 0 ensures that all states i 1 are communicating for Z n and, as a consequence, that all quasi invariant measures see Section 3 have positive mass at each i 1. The function rk =rκ, q, k now clearly takes the form rk = κ q k + E k τ j=1 κ q Zj j 1 1 κ Z i 2.3 for k IN and 0 <κ q<1. Since rk is also a functional of Z τ n 0 n τ under P k, its asymptotic behavior, as k, should be linked to the limit behavior of Z τ n 0 n τ under P k. Unfortunately, there is not just one limiting distribution but infinitely many, essentially the Martin entrance boundary of Z n. This comes out from potential theoretic considerations for subcritical GWPs as described e.g. in [3]. A short review of the most important facts from there will be given in the following section. Here we confine ourselves to a sketchy description in order to formulate our results. Let Q k be the distribution of the time reversal Z τ n 0 n τ under P k. Any probability measure Q in the closure of {Q k,k 1} with respect to weak convergence defines a Markov chain W n on IN 0 with transition matrix q ij i,j 0, say, and corresponds uniquely to a quasi invariant measure η =η i i 1 for Z n see Section 3 via the relation q ij = 0, if i = j =0 η j p j 0, if i =0,j 1, η i p ij, η j if i, j 1, 2.4 where η is normalized such that j 1 η jp j 0 = 1. In our setting, we are interested in sequences k n,n 1,approaching in such a way that rk n converges, as n. It suffices to consider sequences k n,n 1, such that Q kn converges weakly to some probability measure Q. Wemay identify Q with a quasi invariant measure η via 2.4. As shown in [2], these are exactly the extremal elements in the convex set of all quasi invariant measures normalized as above, for which the circle forms a natural parametrization. We thus identify the closure of {Q k,k 1} with the set N def = IN 1, 0]. The Martin topology on N, rendering weak continuity of x Q x, is isomorphic to the topology generated by the metric ρ defined in Section 3. Taking these facts for granted, assertion 2.5 of Theorem 2.1 below should no longer be too surprising.

7 Theorem 2.1. Suppose 2.1, 2.2 and 0 <κ<q. Then, for all x 1, 0], lim rk = rx def = E k x ρ 5 Wn x κ 1 κ W ix W 0 x = q i>n where W n x is a Markov chain on IN 0 with distribution Q x. Moreover, for each q 0, 1, there exist infinitely many κ 0,q such that r is not a constant. Theorem 2.2. Suppose 2.1, 2.2, κ = q, and put def = E k τ. Then lim k qk = qk 3. Quasi invariant measures and time reversal We begin with a review of some basic facts from potential theory for subcritical GWPs as described in Athreya and Ney [3, Chapter II]. The notation is kept from there as far as possible. So let Z n be an ordinary subcritical GWP with reproduction distribution p j j 0 and reproduction mean µ = j 0 jp j < 1. Let f be the generating function of p j j 0, i.e. fs = j 0 p js j and f n its n-fold iterate. Denote by p ij the transition probabilities of Z n.aσ-finite measure η =η j j 1 on IN is called quasi invariant or quasi stationary for Z n if η j = i 1 η i p ij for all j IN. Notice that we exclude the absorbing state 0 in the summation. The generating function Us = j 1 η js j of any such η is analytic for s < 1 and, if normalized so that Up 0 = 1, satisfies the functional relation 1+Us = Ufs. Conversely, this relation characterizes quasi invariant measures [3, Theorem II.2]. Since all states i 1 communicate and η j = i 1 η ip n ij for all n 1, we infer η i > 0 for all i 1, as already mentioned in Section 2. In order to describe all quasi invariant measures for Z n, let [3, II.2, eq. 3] Us, t def = n Z expqsµ n t expq0µ n t, s < 1,t 1,0]. Here Qs def = lim n µ n f n s 1, s [0, 1].

8 6 Note that Qfs = µqs [3], eq. 10 on p. 40] and Qs =Q01 Bs [3, eq. 30 on p. 47], where B is the generating function Bs def = j 1 b j s j, b j = lim n P Z n = j Z n > 0. The following result is shown in [2]. Theorem 3.1. If EZ 1 log Z 1 <, then the space of quasi invariant measures up to positive scalars is isomorphic to the set of probability measures on the circle. The bijection η ν can be stated as U η = U,t νdt 3.1 where U η is the generating function of η. ϕx def = 1,0] Recall that N = IN 1, 0]. Define the map ϕ : N C by and then the metric ρ on N by x 1+x e2πi log µ x, if x IN e 2πix, if x 1, 0] ρx, y = ϕx ϕy. Notice that under this metric the closure of IN is 1, 0] and that 1, 0] is endowed with the spherical topology. The latter is not true for the metric given in [3, p. 69]. An integer sequence k n n 1 converges in the ρ-metric iff Kk n, n 1 converges pointwise on IN 0, where Ki, j def = Gi, j k 1 Gi, kpk 0 is the Martin kernel and Gi, j def = p n ij is the Green kernel. Every such sequence k n with a ρ-limit t 1, 0] will be called a ρ Martin sequence hereafter, and we write k n t equivalent is ϕkn ϕt. The closure of {K,j,j IN} is isomorphic to N,ρ. For such a Martin sequence we further have lim Kk n,j = n lim Gk n,j = η j t, n where ηt =η j t j 1 is the quasi invariant measure with generating function U,t, t 1, 0] as defined above. For the first equality it should be noticed that Gk n,lp l 0 = f k n m p 0 f k n m 0 = f k n m+1 0 f k n m 0 = 1 f k n 0 l 1 m 1 m 1

9 7 which converges to 1 as n. Note also that ηt is continuous in t, see [3, p. 69]. The time reversal W n t,say,ofz n with respect to any quasi invariant measure ηt is a Markov chain with n-step transition matrix Q n t =q n ij t i,j 0, n 1, where q n ij def t = 0, if i = j =0 η j tp j τ=n, if i =0,j η j tp n ji, if i, j 1 η i t The associated Green function is denoted Hi, j, t = qn ij t and satisfies H0, 0,t=1, H0,j,t=η j t for j 1 and Hi, j, t =η j tgj, i/η i t, otherwise. Lemma 3.2. For any i 1,..., i m IN und m IN, the function f i1,...,i m : IN 1, 0] [0, 1], f i1,...,i m t def = { is continuous in the ρ-metric. P t Z τ m = i m,..., Z τ 1 = i 1,Z τ =0, if t IN P W 1 t =i 1,..., W m t =i m W 0 t=0, if t 1, 0] Proof. Let first IN k n ρ t 1, 0] be a Martin sequence. Then, as n, f i1,...,i m k n = P kn Z τ m = i m,..., Z τ 1 = i 1,Z τ =0 = kn Z l =i m p im i m 1... p i1 0 l 0P 1 = Gk n,i m η im t q i m 1 i m t... q 0i1 t q im 1 i m t... q 0i1 t = f i1,...,i m t. For a sequence 1, 0] t n in t. ρ t 1, 0], the assertion follows from the continuity of the ηj t Notice that Lemma 3.2 states in particular that for every Martin sequence k n ρ t, lim n P k n Z τ m = i m,..., Z τ 1 = i 1,Z τ =0 = PW 1 t=i 1,..., W m t =i m W 0 t=0 for all i 1,..., i m IN und m IN which explains the meaning of W n t as a time reversal of Z n.

10 8 4. Proof of Theorem 2.1 Let R : N [0, 1 [0, 1 [0, ] be the function defined through τ n 1 E x u Z n 1 v Z i, if x IN n=0 Rx, u, v = E u W nx v i>n1 Wix W 0x =0, if x 1, 0]. The important fact we will prove in this section is Proposition 4.1. The function R is finite and continuous in the product topology induced by N,ρ [0, 1 2, Euclidean. The proof of this result is provided through a series of lemmata. Fix N IN and define R N x, u, v = E N n=0 u W nx E x τ n=τ N N i=n+1 u Z n n 1 i=τ N 1 v Z i, if x IN. 1 v Wix W 0x =0, if x 1, 0] Our program is to show first that R N is continuous for each N Lemma 4.2 and then in several steps that R R N converges to 0 uniformly on compact sets Lemma This clearly implies the asserted continuity of R. Lemma 4.2. For each N IN, the function R N is continuous in the product topology induced by N,ρ [0, 1 2, Euclidean. Proof. Fix N IN, take a sequence x n,u n,v n convergent to x, u, v and write R N x n,u n,v n R N x, u, v R N x n,u n,v n R N x n,u,v + R N x n,u,v R N x, u, v. The second expression on the right-hand side tends to 0 by an application of Lemma 3.1 because R N,u,v is the expectation of a bounded function w.r.t. the weakly convergent discrete probability distributions P x Z τ N,..., Z τ. As for the first, it is easy to show uniform convergence in x n. Indeed, if u u <εand v v <ε, then R N x, u,v R N x, u, v τ E x u + ε Z n n=τ N = E x ε,u,v Z τ N,..., Z τ n 1 i=τ N 1 v ε Z i u ε Z n n 1 i=τ N 1 v + ε Z i

11 9 where ε,u,v is a bounded function defined in the obvious manner. Notice that ε,u,v 0as ε 0. Hence, by another appeal to Lemma 3.1 and the monotone convergence theorem, lim R Nx n,u n,v n R N x n,u,v n lim lim E x ε 0 n n ε,u,v Z τ N,..., Z τ = lim E ε,u,v W N x,..., W 0 x W 0 x =0 = 0. ε 0 In order to show uniform compact convergence of R N to R, asn, we first observe that for k IN τ N 1 n 1 Rk, u, v R N k, u, v E k u Z n 1 v Z i where n=0 + E k τ n=τ N τ N 1 E k n=0 n 1 u Z n u Z n τ N 1 NE k n=0 i=τ N τ N 1 1 v Z i 1 + NE k 1 u Z n + v Z n τ N 1 2N E k u v Z n 1 n=0 n 1 c i τ N 1 1 v Z i 1 v Z i 4.1 n c i 4.2 for c 0,..., c n [0, 1] has been used for the penultimate inequality. In view of the subsequent estimations we note that 4.2 remains true if n =. Forx 1,0], we further have Rx, u, v R N x, u, v E u W nx v n>n i>n1 Wix W 0x =0 + N E n=0 u W nx N i=n+1 E u Wnx W 0 x =0 n>n 2N E u v W nx W 0 x =0. n>n 1 v Wix 1 1 v Wjx W 0x =0 4.3 j>n + NE 1 1 v Wjx W 0x =0 j>n Since that latter inequality is easier to handle we first show

12 10 Lemma 4.3. For all y, w < 1, lim sup N x y;u,v w Rx, u, v R N x, u, v = 0. Proof. Recalling from 3.2 the definition of q n 0i, we obtain N n>ne u v W nx W 0 x =0 N Ew Wnx W 0 x =0 n>n = N w i q n 0i n>n i 1 = N w i η i xp i τ = n n>n i 1 = N w i η i xp i τ>n i 1 = N i 1w i η i xp i Z N >0 = N i 1 w i η i x1 f i N 0 = N Uw, x Uwf N 0,x Cw, ynw1 f N 0, where Cw, y def = max x y;u w D u Uu, x < as one can easily check. The assertion now follows because N1 f N 0 0asN, see [3, Section I.11]. In order to further exploit 4.1 for our purposes, we have to consider the functions τ N 1 g N k, u def = E k n=0 u Z n, hk, u def = E k u Z n 1 {Zn >0} = fnu k fn0 k, k, N IN, u [0, 1, which are related as follows: g N k, u = E k u Z n 1 {τ>n+n} = E k u Z n 1 {Zn+N >0} = E k u Z n P Zn Z N > 0

13 11 = E k u Z n 1 f Z n N 0 = fnu k fnuf k N 0 = hk, u hk, uf N Lemma 4.4. The function h satisfies sup hk, u mu 4.5 k IN for each u<1, where mu def = inf{n 1:f n 0 u}. Furthermore, N w sup hk, u hk, v < mw+ε 2 + u v k 1;u,v w w+ε w N 4.6 for all ε>0,0<w<1 εand N IN. Proof. By using 0 1 f nu k < for all k IN and u<1 we obtain with m = mu hk, u = = fnu 1 k fnu k fnf k m 0 m 1 fn mf k m 0 1 n m n=0 m fn0 k m n=0 because the first sum in the previous line is negative. In order to prove 4.6, we note first that fn0 k 1 hk, u = E k u Z n 1 {Zn >0} = i 1 u i P k Z n = i = i 1u i Gk, i. Define h N k, u def = N i=1 ui Gk, i and choose an arbitrary ε>0, w.l.o.g. < 1 w. Then, for all k IN and u w hk, u h N k, u = i>n u i Gk, i = N u w + ε i Gk, i w + ε i>n N N u w hk, w + ε mw + ε, w + ε w + ε 4.7

14 where 4.5 has been used for the final inequality. Moreover, for all u, v w and N 1, 12 h N k, u h N k, v = u v = u v u v w N u i v i i=1 u v Gk, i N i 1 u j v i 1 j Gk, i i=1 j=0 N iw i 1 Gk, i i=1 u v w Nmw, 4.8 the last inequality again by 4.5. By combining 4.7 and 4.8 with a simple application of the triangle inequality, we finally obtain 4.6 Going back into 4.1, we are now ready to prove Lemma 4.5. For all w<1, lim sup N k 1;u,v w Rk, u, v R N k, u, v = 0. Proof. Indeed, we infer with the help of 4.1, 4.4 and the previous lemma lim sup sup N k 1;u,v w 2 lim sup N N Rk, u, v R N k, u, v sup k 1;u,v w 2mw + ε lim sup N 2N hk, u v hk, u vf N 0 N w + u v1 f N0 N 2 w + ε w = 0 recalling that 1 f N 0 converges to 0 exponentially fast [3, I.11]. Proof of Proposition 4.1. A combination of Lemma 4.3 and 4.5 clearly yields uniform compact convergence of the R N to R. Since the R N are further continuous by Lemma 4.2, we conclude the continuity of R as claimed. In view of the main assertion of Theorem 2.1, namely the nonconstancy of the extinction probability ratio rx for suitable pairs κ, q, two further lemmata are needed. Lemma 4.6. The function η 1 t, t 1, 0], is not a constant. Proof. Note first that η 1 t = D s Us, t s=0 = Q 0 n Z µ n t exp Q0µ n t.

15 13 Since our assumptions in Section 2 guarantee η 1 to be everywhere positive, we particularly have Q 0 > 0. We make the change of variables x = µ t, i.e. t = log µ x. Defining Ψx, y def = e xyµn e yµn, y < 0 <x, n Z we obviously have D x Ψx, y = n Z yµ n e xyµn and therefore η 1 log µ x = Q 0 Q0 xd xψx, Q0. Now suppose η 1 be constant and infer D x Ψx, Q0 = c x for all x [1, 1/µ and some constant c<0. The equality extends to all x>0 because both sides are evidently analytic functions on the half plane of complex numbers with positive real part. Integration together with Ψ1, 0 then implies Ψx, Q0 = x 1 D z Ψz,Q0 dz = c log x 4.9 for all x>0. Next, the functional equation xy Ψx, y = Ψ z,z for all y, z < 0 <xtogether with 4.9 leads to xy Q0 Ψx, y = Ψ Q0,Q0 + Ψ y,y z + Ψ y,y xy = clog Q0 Q0 + Ψ y,y for all y<0<x.forx=1,weget Q0 Ψ y,y Q0 = clog y and thereby Ψx, y = clog x for all y<0<x. Rewriting this result for Us, t, we find that Qs Us, t = Ψ Q0,µ t Q0 = c log Qs Q0 which is impossible because the U,t are pairwise distinct by Theorem 3.1. Lemma 4.7. Given any q 0, 1, the function R, u/q, u is not constant for all sufficiently small u 0,q.,

16 Proof. Rx, u/q, u is analytic in u and R, 0, 0 1. Writing 14 Rx, u/q, u = E 1 u Wix W 0x =0 i 1 + u q E n 1 Wn x 1 u 1 u Wix q W 0x =0 i>n and noting that W n x 1 for all x 1, 0] and n 1, it is easily verified that D u Rx, u/q, u u=0 = 1 q η 1 x. q Since η 1 x is not constant in x, the same holds true for D u Rx, u/q, u atu= 0. Consequently, picking two distinct values x 1,x 2 1,0] with D u Rx 1, u/q, u u=0 D u Rx 2, u/q, u u=0, we must also have Rx 1, u/q, u Rx 2, u/q, u for all sufficiently small u 0,q using Rx 1, 0, 0 = Rx 2, 0, 0 and the continuity of D u Rx, u/q, u inu. This proves the lemma. Proof of Theorem 2.1. Suppose 0 <κ<q. Since rk =κ/q k + Rk, κ/q, κ for k IN see 2.3 and rx =Rx, κ/q, κ for x 1, 0], assertion 2.5 follows directly from Proposition 4.1. Moreover, we infer from the previous lemma that r is not a constant for infinitely many, in fact all sufficiently small κ 0,q. We have thus proved the theorem. 5. Proof of Theorem 2.2 We begin with an auxiliary lemma which gives an asymptotic estimate of the expected extinction time = E k τ as k. Lemma 5.1. Let Z n be a subcritical GWP with reproduction mean µ>0and EZ 1 log Z 1 <. Then lim k log 1/µ k = 1. Proof. Recall from [3, I.11] that f n 0=1 c n µ n with positive constants c n [0, 1] converging to some c>0. It is the positivity of c where the X log X-condition enters. Since P k τ>n=p k Z n >0=1 fn0, k we infer log 1/µ k = 1 log 1/µ k = 1 log 1/µ k P k τ>n = 1 log 1/µ k 1 1 c n µ n k 1 fn0 k Fix any ε 0, 1, put n = n ε, k def =1 ε log 1/µ k, n = n ε, k def =1+ε log 1/µ k and split up the sum into three parts S 1 k,s 2 k,s 3 k ranging from 0 to n 1, from n to n 1,

17 15 and from n to, respectively. Note that µ n = k 1 ε and µ n = k 1+ε. The three sums will be considered separately. Choose m such that inf n m c n c/2. Then we have for S 1 k m 1 ε 1 1 cµ m /2 k log 1/µ k 1 log 1/µ k S 1 k n 1 n=m 1 1 cµ n /2 k n µ n k log 1/µ k n=0 1 ε 1 1 µ n k = 1 ε 1 1 k 1 ε k 1 ε 1 exp 2k ε, where the last inequality holds for sufficiently large k using log1 x 2xfor all positive x sufficiently close to 0. Consequently, lim k S 1k = 1 ε. For S 2 k we just note 0 S 2 k 2ε. Finally, we obtain for S 3 k, if k is sufficiently large, 0 S 3 k = 1 log 1/µ k 1 log 1/µ k 1 log 1/µ k 1 1 µ n +n k 1 1 k 1+ε µ n k 2 k ε log 1/µ k 1 exp 2k ε µ n where 1 e x x for all x has been used for the final inequality. Hence µ n lim k S 3k = 0. Putting the results together the assertion of the lemma easily follows because ε 0, 1 was arbitrary. Proof of Theorem 2.2. We first note that τ qk q k = rk = 1 1+E k j=1 j 1 1 q Z i

18 because κ = q. We thus have to show lim k 1 E k τ 1 j=1 16 j 1 1 q Z i = 0. Fix an arbitrary ε 0, 1, put Nk def = ε and split up the sum under the expectation into the sum from 1 to τ Nk 1 of course, equal to 0 if τ Nk 0 and the sum from τ Nk toτ. As for the latter, we immediately have 0 lim sup k 1 E k τ j=τ Nk 1 j 1 1 q i Z Nk+1 lim = ε. k Turning to the first sum, we use once more the inequality 1 n 1 x i n x i for numbers x 1,..., x n [0, 1] and obtain τ Nk 1 1 E k j=1 1 1 τ Nk 1 E k j=1 j 1 1 q Z i j 1 q Z i 1 j 1 E k q Z i 1 {Zj+Nk >0} j 1 = 1 E k q Z i P Zi Z j i+nk > 0 = 1 = 1 i 0 j>i i 0 j>nk i 0 j>nk E k q Z i 1 f Z i j 0 fi k q fi k qf j 0. Now a first order Taylor expansion of f k i qf j0 about q together with the monotonicity of f i and f i gives for some z ij between qf j 0 and q fi k qf j 0 = fi k q kqf k 1 i z ij f iz ij 1 f j 0 fi k q kqf k 1 i qf i11 f j 0. Hence the above estimation can be continued as kq f k 1 i 0f i1 kq i 0 i 0 f k 1 i qµ i j>nk j>nk 1 f j 0 2cµ j. 5.1

19 17 Now the second sum in 5.1 is clearly bounded by a constant times µ ε log 1/µ k = k ε for all k since Nk = ε εlog 1/µ k by Lemma 5.1, while the first can be bounded by a constant times k 1 ε for sufficiently large k. To see the latter, split up the first sum as 1 ε log 1/µ k + f k 1 i qµ i. Observe that i> 1 ε log 1/µ k i> 1 ε log 1/µ k f k 1 i qµ i µ1 ε log 1/µ k 1 µ = k 1 ε 1 µ. Since, for all i 0 i 1 ε log 1/µ k, i 0 sufficiently large independent of k, all k sufficiently large and some Qq 1, 0, see [3, I.11], we further have f k 1 i q 1 + Qqµ i /2 k 1 1 ε log 1/µ k 1 + Qqµ 1 ε log 1/µ k /2 k 1 expk 1 ε Qq/2, f k 1 i qµ i i 0 f k 1 q+1 ε log 1/µ k expk 1 ε Qq/2 for all k sufficiently large. Putting the pieces together, we finally conclude in 5.1 kq f k 1 i qµ i 2cµ j const i 0 which converges to 0 as k. i 0 j>nk

20 18 References [1] ALSMEYER, G. and RÖSLER, U The bisexual Galton-Watson process with promiscuous mating: Extinction probabilities in the supercritical case. Ann. Appl. Probab. 6, [2] ALSMEYER, G. and RÖSLER, U The Martin entrance boundary of the Galton- Watson process. In preparation [3] ATHREYA, K.B. and NEY, P Branching Processes. Springer, New York. [4] BIGGINS, J.D. and NADARAJAH, S Near-constancy of the Harris function in the simple branching process. Comm. Statist. - Stoch. Models 9, [5] BRUSS, T A note on extinction criteria for bisexual Galton-Watson processes. J. Appl. Probab. 21, [6] DALEY, D Extinction probabilities for certain bisexual Galton-Watson branching processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 9, [7] DALEY, D., HULL, D.M. and TAYLOR, J.M Bisexual Galton-Watson branching processes with superadditive mating functions. J. Appl. Probab. 23, [8] HULL, D.M A necessary condition for extinction in those bisexual Galton-Watson branching processes governed by superadditive mating functions. J. Appl. Probab. 19, [9] HULL, D.M Conditions for extinction in certain bisexual Galton-Watson branching processes. J. Appl. Probab. 21,