A Conceptual Proof of the Kesten-Stigum Theorem for Multi-type Branching Processes

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1 Classical and Modern Branching Processes, Springer, New Yor, 997, pp Version of 7 Sep A Conceptual Proof of the Kesten-Stigum Theorem for Multi-type Branching Processes by Thomas G. Kurtz, Russell Lyons, Robin Pemantle, and Yuval Peres Abstract. We give complete proofs of the theorem of convergence of types and the Kesten-Stigum theorem for multi-type branching processes. Very little analysis is used beyond the strong law of large numbers and some basic measure theory. Consider a multi-type Galton-Watson branching process with J types. Let L i,j) be a random variable representing the number of particles of type j produced by one type-i particle in one generation. For :=,..., J ), let p i) = P[ j Li,j) = j ]. Assume that m i,j) = E[L i,j) ] is finite for all pairs i,j). For any J-vector vector x = x,...,x J ), write x := x + + x J. Let ρ be the maximum eigenvalue of the mean matrix M := m i,j) ) with left unit eigenvector b, where unit means that b =. We assume that the process is supercritical i.e., ρ > ) and positive regular i.e., some power of M has all entries positive). Let Z n j) be the number of particles of type j in generation n and Z n := Z n ),...,Z n J) ). All vectors are row vectors unless otherwise specified. The Kesten- Stigum theorem says the following Kesten and Stigum 966), Athreya and Ney 972), p. 92): Theorem. There is a scalar random variable W such that ) and P[W > 0] > 0 iff 2) E J i,j= Z n = Wb ρn L i,j) log + L i,j) <. 99 Mathematics Subject Classification. Primary 60J80. Key words and phrases. Galton-Watson, size-biased distribution. Research partially supported by two Alfred P. Sloan Foundation Research Fellowships Lyons and Pemantle), and NSF Grants DMS Lyons), DMS Pemantle), and DMS Peres).

2 We shall give a proof of this theorem that avoids much analysis, extending the proof of the single-type case given in Lyons, Pemantle and Peres 995). The multi-type case has an additional difficulty not present in the single-type case: namely, the convergence of the quotient in ) is no longer automatic. Thus, we begin with an elementary proof of this result simplifying Kurtz 973). Theorem 2. Convergence of Types) Almost surely on nonextinction, we have Z n Z n = b. Note that no moment assumptions beyond finite means are made. We shall use the following elementary consequence of the strong law of large numbers. { } Lemma 3. Suppose that N are random variables and that X n ) ; n, are i.i.d. mean-zero random variables. On the event { d N ; inf N +d /N > }, we have N N n= X ) n = 0 We also need the following lemma. Lemma 4. Suppose that {L n) i ; n,i } are i.i.d. random variables with values in N and with mean m >. If {V n } are N-valued random variables such that V n+ V n i= Ln) i for all n, then inf V n+ /V n m on the event E := { V n 0}. Proof. By comparison with a single-type branching process, it follows that V n grow exponentially on E. Choose any m < m. By truncating the random variables L n) i to a level with mean larger than m, we see by Chebyshev s inequality that there is some constant C such that P[V n+ < m V n V n ] C/V n for all n. The conditional Borel-Cantelli lemma Durrett 99), p. 207) then implies that on the event that V n grows exponentially, inf V n+ /V n m. Since this event occurs when E does and m is arbitrary, the result follows. Proof of Theorem 2. Let L i,j) n, be the number of type j children of the th type-i particle in generation n, so that for all n 0 and j J, Z j) n+ = J i= = L i,j) n,. Because the process is supercritical and positive regular, for each i, there is some d N such that for each, the variables { dn+ ; n 0} dominate a single-type supercritical 2

3 branching process. Therefore, Lemma 4 shows that the event in Lemma 3 occurs on nonextinction. Hence we may apply Lemma 3 to obtain that for each i,j), n = L i,j) n, mi,j)) = 0 Taing a weighted average of these equations, we see that for each j, Z j) Z n n+ J i= ) Z n i) m i,j) = Z n J i= = L i,j) n, mi,j)) = 0 For simplicity, write v n := Z n / Z n, A := M/ρ, and γ n+ := Z n+ /ρ Z n ). Then γ n+v n+ v n A = 0 Since v n γ n j v n A = γ n r v n r v n r A)A r r=0 the triangle inequality yields that for every, j=r+ γ n j, v n γ n j v n A = 0 But A cb, where c is a right column ρ-eigenvector. Choosing large enough, we can therefore mae arbitrarily small, which means sup sup v n v n γ n j v n cb v n c/ γ n j b can also be made arbitrarily small. Since v n and b are unit vectors, this implies that v n b Proof of Theorem. Let c be a right column ρ-eigenvector. For any tree t with J possible types of vertices, define W n t) := ρ nz nt)c Z 0 t)c. 3

4 For r =,...,J, let GW r) denote multi-type Galton-Watson measure with one initial particle of type r. Then it is easily seen and well nown that W n is a GW r) -martingale. We shall show that its it is non-degenerate iff 2) holds. We first construct some useful measures on trees. Set p i) := pi) c ρc i. Given r 0 [, J], start with one particle v 0 of type r 0. Generate offspring according to the probabilities p r 0). Pic one of these children v at random, with children being piced with probabilities proportional to c j when their type is j. The children other than v get ordinary independent GW j) trees, while v gets an independent number of offspring according to the probabilities p r ), where r is the type of v. Again, pic one of the children of v at random, call it v 2, and give the others ordinary independent GW j) trees, and so on. Define the measure ĜW r 0) as the joint distribution of the random tree and the random path v 0,v,v 2,...). Let its marginal on the space of trees be ĜW r 0). For any rooted tree t and any n 0, denote by [t] n the set of rooted trees whose first n levels agree with those of t. In particular, if the height of t is less than n, then [t] n = {t}.) If v is a vertex at the nth level of t, then let [t;v] n denote the set of trees with distinguished paths such that the tree is in [t] n and the path starts from the root, does not bactrac, and goes through v. Assume that t is a tree of height at least n + and that the root of t is of type r and has children with descendant trees t ),t 2),...,t ) having roots of types r,...r. Any vertex v in level n + of t is in one of these, say t i). The measures ĜW r) clearly satisfy the recursion ĜW r) [t;v] n+ = p r) ci i) cĝwr [t i) ;v] n GW rj) [t j) ] n By induction, we conclude that = pr) j i ρ ĜW r i) [t i) ;v] n GW rj) [t j) ] n. ĜW r) [t;v] n = j i c i ρ n Z 0 t)c GWr) [t] n for all n and all [t;v] n as above, where v is of type i. Therefore, 3) ĜW r) [t] n = W n t)gw r) [t] n 4

5 for all n and all trees t. Now 2) is equivalent to 4) J j= p j) log+ <. The remaining details of the proof are a straightforward modification of the proof for the single-type case given in Lyons, Pemantle and Peres 995). Namely, by conditioning on the numbers of children of the vertices v n, one shows that with respect to the measure ĜW r 0), we have that supw n < is equivalent to 4). On the other hand, the Radon-Niodym relation 3) shows that supw n < ĜW r0) - is equivalent to W n > 0 with positive GW r 0) -probability. REFERENCES Athreya, K. B. and Ney, P. 972). Branching Processes. Springer, New Yor. Durrett, R. 99). Probability: Theory and Examples. Wadsworth, Pacific Grove, California. Kesten, H. and Stigum, B. 966). A it theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37, Kurtz, T. G. 973). Almost sure convergence of the type distribution for a supercritical branching process, unpublished manuscript. Lyons, R., Pemantle, R. and Peres, Y. 995). Conceptual proofs of L log L criteria for mean behavior of branching processes, Ann. Probab. 23, Department of Mathematics, University of Wisconsin, Madison, WI Department of Mathematics, Indiana University, Bloomington, IN Department of Mathematics, University of Wisconsin, Madison, WI Department of Statistics, University of California, Bereley, CA