Chapter 2 Galton Watson Trees

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1 Chapter 2 Galton Watson Trees We recall a few eleentary properties of supercritical Galton Watson trees, and introduce the notion of size-biased trees. As an application, we give in Sect. 2.3 the beautiful conceptual proof by Lyons et al. [76] of the Kesten Stigu theore for the branching process. The goalof this briefchapteristo givean avant-goût of the spinal decoposition theore, in the siple setting of the Galton Watson tree. If you are already failiar with any for of the spinal decoposition theore, this chapter can be skipped. 2. The Extinction Probability Consider a Galton Watson process, also referred to as a Bienayé Galton Watson process, with each particle (or: individual) having i children with probability p i (for i 0; P jd0 p j D ), starting with one initial ancestor. To avoid trivial discussions, we assue throughout that p 0 C p <. Let Z n denote the nuber of particles in the n-th generation. By definition, if Z n D 0 for a certain n,thenz j D 0 for all j n. We write q WD PfZ n D 0 eventuallyg; X WD E.Z / D ip i 2.0; : id0 (extinction probability) (ean nuber of offspring of each individual) Theore 2. (i) The extinction probability q is the sallest root of the equation f.s/ D sfor s 2 Œ0;, wheref.s/ WD P id0 si p i, 0 0 WD. (ii) In particular, q D if, and q <if <. Springer International Publishing Switzerland 205 Z. Shi, Branching Rando Walks, Lecture Notes in Matheatics 25, DOI 0.007/ _2

2 2 2 Galton Watson Trees f (s) f (s) Case > Case p 0 p 0 0 q < s 0 q = s Fig. 2. Generating function of the reproduction law Proof By definition, f.s/ D E.s Z /,ande.s Z n j Z n / D f.s/ Z n.soe.s Z n / D E. f.s/ Z n /, which leads to E.s Z n / D f n.s/ for any n,wheref n denotes the n-th fold coposition of f. In particular, P.Z n D 0/ D f n.0/. Since fz n D 0g fz` D 0g for all n `,wehave [ q D P fz n D 0g n D li n! P.Z n D 0/ D li n! f n.0/: The function f W Œ0;! R is increasing and strictly convex, with f.0/ D p 0 0 and f./ D. It has at ost two fixed points. Note that D f 0. /.SeeFig.2.. If, thenp 0 >0,andf.s/ >s for all s 2 Œ0; /. Sof n.0/!. Inother words, q D is the unique root of f.s/ D s. If 2.;, thenf n.0/ converges increasingly to the unique root of f.s/ D s, s 2 Œ0; /. In particular, q <. ut It follows that in the subcritical case (i.e., <) and in the critical case ( D ), there is extinction with probability, whereas in the supercritical case ( >), the syste survives with positive probability. If <, we can define M n WD Z n n ; n 0: Since.M n / is a non-negative artingale with respect to the natural filtration of.z n /, we have M n! M a.s., where M is a non-negative rando variable. By Fatou s lea, E.M / li inf n! E.M n / D. It is, however, possible that M D 0. So it is iportant to know whether P.M >0/is positive. If there is extinction, then trivially M D 0. In particular, by Theore 2., we have M D 0 a.s. if. What happens if >? Lea 2.2 Assue <. ThenP.M D 0/ is either q or. Proof We already know that M D 0 a.s. if. So let us assue < <.

3 2.2 Size-Biased Galton Watson Trees 3 By definition, Z nc D P Z id Z.i/ n (notation: P WD 0), where Z.i/ n, i, are copies of Z n, independent of each other and of Z. Dividing both sides by n and letting n!, it follows that M has the law of P Z id M.i/, wherem,.i/ i, are copies of M, independent of each other and of Z. Hence P.M D 0/ D EŒP.M D 0/ Z D f.p.m D 0//, i.e., P.M D 0/ is a root of f.s/ D s, so P.M D 0/ D q or. ut Theore 2.3 (Kesten and Stigu [55]) Assue < <.Then E.M / D, P.M >0j non-extinction/ D, E.Z ln C Z /<; where ln C x WD ln axfx; g. Theore 2.3 says that E.M / D, P.M D 0/ D q, P id p i i ln i <. The proof of Theore 2.3 is postponed to Sect We will see that the condition E.Z ln C Z /<, apparently technical, is quite natural. 2.2 Size-Biased Galton Watson Trees In order to introduce size-biased Galton Watson trees, let us view the tree as a rando eleent in a probability space. ; F ; P/, using the standard foralis. Let U WD f g[ S kd.n / k,wheren WD f; 2;:::g. For eleents u and v of U,letuv be the concatenated eleent, with u D u D u. Atree! is a subset of U satisfying the following properties: (i) 2!; (ii) if uj 2! for soe j 2 N,thenu 2!; (iii) if u 2!, thenuj 2! if and only if j N u.!/ for soe non-negative integer N u.!/. In words, N u.!/ is the nuber of children of the vertex u. Vertices of! are labeled by their line of descent: the vertex u D i :::i n 2 U stands for the i n - th child of the i n -th child of ::: of the i -th child of the initial ancestor. See Fig Fig. 2.2 Verticesofatreeas eleents of U

4 4 2 Galton Watson Trees Let be the space of all trees, endowed with a -field F defined as follows. For u 2 U,let u WD f! 2 W u 2!g be the subspace of consisting of all the trees containing u as a vertex. [In particular, D.] Let F WD f u; u 2 U g. Let T W! be the identity application. Let. p k ; k 0/ be a probability, i.e., p k 0 for all k 0, and P kd0 p k D. There exists a probability P on. ; F / [203] such that the law of T under P is the law of the Galton Watson tree with reproduction distribution. p k /. Let F n WD f u; u 2 U ; juj ng, wherejuj is the length of u (or the generation of the vertex u in the language of trees). Note that F is the sallest -field containing all the F n. For any tree! 2,letZ n.!/ be the nuber of individuals in the n-th generation, i.e., Z n.!/ WD #fu 2 U W u 2!; juj Dng. It is easily checked that for any n, Z n is a rando variable taking values in N WD f0; ; 2; : : :g. Assue now <. Since.M n / is a non-negative artingale, we can define Q to be the probability on. ; F / such that for any n, Q jfn D M n P jfn ; where P jfn and Q jfn are the restrictions of P and Q on F n, respectively. For any n, Q.Z n >0/D EŒ fzn >0gM n D EŒM n D, which yields Q.Z n > 0; 8n/ D : there is alost sure non-extinction of the Galton Watson tree T under the new probability Q. The Galton Watson tree T under Q is called a size-biased Galton Watson tree. We intend to give a description of its paths. We start with a lea. Let N WD N.IfN, we write T, T 2 ;:::, T N for the N subtrees rooted at each of the N individuals in the first generation. Lea 2.4 Let k.ifa,a 2 ;:::,A k are eleents of F,then Q.N D k; T 2 A ;:::; T k 2 A k / D kp k k kx P.A / P.A i /Q.A i /P.A ic / P.A k /: (2.) id Proof By the onotone class theore, we ay assue, without loss of generality, that A, A 2 ;:::, A k are eleents of F n,forsoen. Write Q (2.) for Q.N D k; T 2 A ;:::; T k 2 A k /.Then ZnC Q (2.) D E nc fndk; T 2A ;:::; T k 2A k g : On fn D kg, we can write Z nc D P k id Z.i/ n,wherez n.i/ is the nuber of individuals in the n-th generation of the subtree rooted at the i-th individual in the first generation. Hence Q (2.) D kx P.N D k/ nc id n E Z.i/ n ft 2A ;:::; T k 2A k g o ˇ N D k :

5 2.2 Size-Biased Galton Watson Trees 5 We have P.N D k/ D p k,and EfZ.i/ n ft 2A ;:::; T k 2A k g j N D kg DEŒZ n ft2ai g Y j6di P.A j /; which is n Q.A i / Q j6di P.A j/. The lea is proved. ut It follows fro Lea 2.4 that the root of the size-biased Galton Watson tree has the biased distribution, i.e., having k children with probability kp k ; aong the individuals in the first generation, one of the is chosen randoly according to the unifor distribution: the subtree rooted at this vertex is a size-biased Galton Watson tree, whereas the subtrees rooted at all other vertices in the first generation are usual Galton Watson trees, and all these subtrees are independent. We iterated the procedure, and obtain a decoposition of the size-biased Galton Watson tree into an (infinite) spine and i.i.d. copies of the usual Galton Watson tree: The root DW w 0 has the biased distribution, i.e., having k children with probability kp k. Aong the children of the root, one of the is chosen randoly according to the unifor distribution, as the eleent of the spine in the first generation; let us denote this eleent by w. We attach subtrees rooted at all other children; they are independent copies of the usual Galton Watson tree. The vertex w has the biased distribution. Aong the children of w, we choose at rando one of the as the eleent of the spine in the second generation, denoted by w 2. Independent copies of the usual Galton Watson tree are attached as subtrees rooted at all other children of w, whereas w 2 has the biased distribution. The syste goes on indefinitely. See Fig Having the application of the next section in ind, let us connect the size-biased Galton Watson tree to the branching process with iigration. The latter starts with w 0 = w w 2 w 3 Fig. 2.3 A size-biased Galton Watson tree

6 6 2 Galton Watson Trees no individual (say), and is governed by a reproduction law and an iigration law. At generation n (for n ), Y n new individuals are added into the syste, while all individuals regenerate independently and following the sae reproduction law; we assue that.y n ; n / is a collection of i.i.d. rando variables following the sae iigration law, and independent of everything else up to that generation. The size-biased Galton Watson tree tells us that.z n ; n 0/ under Q is a branching process with iigration, whose iigration law is that of bn, with P.bN D k/ WD kp k,fork. 2.3 Application: The Kesten Stigu Theore We start with a dichotoy theore for branching processes with iigration. Theore 2.5 (Seneta [29]) Let Z n be the nuber of individuals in the n-th generation of a branching process with iigration.y n /. Assue that < <, where denotes the expectation of the reproduction law. Z (i) If E.ln C Y /<,thenli n n! exists and is finite alost surely. n Z (ii) If E.ln C Y / D,thenli sup n n! D, a.s. n Proof (ii) Assue E.ln C Y / D. By the Borel Cantelli lea [02, Theore 2.5.9], li sup n ln Y n! Da.s. Since Z n n Y n, it follows that for any c >, Z li sup n n! D,a.s. c n ln (i) Assue now E.ln C Y /<. By the law of large nubers, li C Y n n! D 0 n a.s., so for any c >0, P Y k k < a.s. c k Let Y be the -field generated by.y n /. Clearly, E.Z nc j F n ; Y / D Z n C Y nc Z n ; thus. Z n / is a subartingale (conditionally on Y ), and E. Z n n j Y / D P n Y k n kd0.in k particular, on the set f P kd0 < g,wehavesup n E. Z n Z j Y /<, so li n n n! n Y k k exists and is finite. Since P. P kd0 Y k k < / D, the result follows. ut We recall an eleentary result [02, Theore 5.3.3]. Let.F n / be a filtration, and let F be the sallest -field containing all F n.letp and Q be probabilities on. ; F /. Assue that for any n, Q jfn P jfn.let n WD dq j Fn dp jfn,andlet WD li sup n! n which is P-a.s.finite.Then Q.A/ D E. A / C Q.A \f Dg/; 8A 2 F :

7 2.4 Notes 7 It follows easily that Q P, <; Q-a.s., E./ D ; (2.2) Q? P, D; Q-a.s., E./ D 0: (2.3) Proof of Theore 2.3 If P id p i i ln i <, thene.ln C bn/ <. By Theore 2.5, li n! M n exists Q-a.s. and is finite Q-a.s. In view of (2.2), this eans E.M / D ; in particular, P.M D 0/ <, thus P.M D 0/ D q (Lea 2.2). If P id p i i ln i D,thenE.ln C bn/ D. By Theore 2.5, li n! M n exists Q-a.s. and is infinite Q-a.s. Hence E.M / D 0 (by (2.3)), i.e., P.M D 0/ D. ut 2.4 Notes The aterial of this chapter is borrowed fro Lyons et al. [76], and the presentation adapted fro Chap. of y lecture notes [22]. Section 2. collects a few eleentary properties of Galton Watson processes. For ore detailed discussions, we refer to the books by Asussen and Hering [3], Athreya and Ney [32], Harris [22]. The foralis described in Sect. 2.2 is due to Neveu [203]; the idea of viewing Galton Watson trees as tree-valued rando variables finds its root in Harris [22]. The technique of size-biased Galton Watson trees, which goes back at least to Kahane and Peyrière [52], has been used by several authors in various contexts. Its presentation in Sect. 2.2, as well as its use to prove the Kesten Stigu theore, coes fro Lyons et al. [76]. Size-biased Galton Watson trees can actually be exploited to prove the corresponding results of the Kesten Stigu theore in the critical and subcritical cases. See [76] for ore details. Seneta s dichotoy theore for branching processes with iigration (Theore 2.5) was discovered by Seneta [29]; its short proof presented in Sect. 2.3 is borrowed fro Asussen and Hering [3, pp. 50 5].

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