An extension of Hawkes theorem on the Hausdorff dimension of a Galton Watson tree

Transcription

1 Probab. Theory Relat. Fields 116, (2000) c Springer-Verlag 2000 Steven P. Lalley Thomas Sellke An extension of Hawkes theorem on the Hausdorff dimension of a Galton Watson tree Received: 30 June 1998 / Revised: 4 February 1999 Abstract. Let T be the genealogical tree of a supercritical multitype Galton Watson process, and let be the limit set of T, i.e., the set of all infinite self-avoiding paths (called ends) through T that begin at a vertex of the first generation. The limit set is endowed with the metric d(ζ,ξ) = 2 n where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence (ζ) of types of the vertices of ζ. Let be the space of all such sequences. For any ergodic, shift-invariant probability measure µ on, define µ to be the set of all µ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency µ( (v)), where (v) is the set of all ω that begin with the word v. Then the Hausdorff dimension of 1 ( µ ) in the metric d is (h(µ) + log q(ω 0,ω 1 )dµ(ω)) + / log 2, almost surely on the event of nonextinction, where h(µ) is the entropy of the measure µ and q(i,j) is the mean number of type-j offspring of a type-i individual. This extends a theorem of Hawkes [5], which shows that the Hausdorff dimension of the entire boundary at infinity is log 2 α, where α is the Malthusian parameter. 1. Introduction Suppose that the individuals of a simple supercritical Galton Watson process are randomly assigned labels from a finite set L. On the event of nonextinction, there will be infinite descent lines (sequences ξ = ξ 1 ξ 2 of individuals such that each ξ n+1 is an offspring of ξ n, and such that ξ 1 is a member of the first generation). Define the limit set of the Galton Watson process to be the set of all infinite descent lines ξ; equivalently, is the space of ends of the genealogical tree of the Galton-Watson process. For each infinite line of descent ξ, define its pedigree (ξ) = l 1 l 2 to be the infinite sequence of labels asssigned to the individuals ξ 1,ξ 2,... of ξ. A subset B of = L is called non-polar if there is positive probability that B ( ). Characterizations of non-polar sets B have been given in [3, 8, 9]. S.P. Lalley: University of Chicago, Department of Statistics, Chicago, IL 60637, USA. lalley@stat.purdue.edu T. Sellke: Purdue University, Department of Statistics, West Lafayette, IN 47907, USA. tsellke@stat.purdue.edu Supported by NSF Grant DMS Mathematics Subject Classification (1991): 60J80

2 42 S. P. Lalley, T. Sellke This paper concerns the size of 1 (B) for certain non-polar sets B. The sets of primary interest are the sets µ consisting of all µ-generic sequences, where µ is an ergodic, shift-invariant probability measure on. A sequence ω is called µ-generic if every finite sequence x = x 1 x 2 x n with entries in L occurs with limiting frequency µ( (x)) in ω, where (x) is the set of all ω whose first n entries coincide with the entries x i of x. The natural way to measure the size of 1 ( µ ) is by Hausdorff dimension. The idea of using Hausdorff dimension as a measure of size for a Galton Watson tree (or its space of ends) derives from a paper of Hawkes [5] (see also [4] and [8]). Let d = d and d = d be the metrics on and defined by d(ξ,ζ)) = 2 n(ξ,ζ ), (1) respectively, where n(ξ, ζ ) is the smallest integer n such that ξ n ζ n.hawkes proved the following theorem: Hawkes theorem. If the offspring distribution has mean α>1and finite second moment then, almost surely on the event of nonextinction, the limit set of the Galton Watson tree has Hausdorff dimension (in the metric d ) δ H ( ) = log α log 2. (2) R. Lyons [8] subsequently showed that the second moment hypothesis is unecessary. The main result of this paper is an explicit formula for the Hausdorff dimension of 1 ( µ ) in the metric d defined by (1). Apart from its intrinsic interest, this result is crucial for the main results of [6] concerning the backscattering phenomenon for anisotropic branching random walks on homogeneous trees. The formula holds generally for a multi-type Galton Watson process G (see [1], Ch. V for background) with finite type space L. For each pair i, j L let q ij = q(i,j) be the mean number of type-j offspring produced by a type-i individual, and let Q = (q ij ) i,j L be the L L matrix of means. We shall make the following assumptions: (H1) The mean offspring numbers q ij are all finite. (H2) The matrix Q is irreducible; equivalently, some positive power Q n of Q has strictly positive entries. For simplicity, we shall assume that the 0th generation consists of a single individual of type i, where i is a distinguished element of L. Thus, the genealogical tree T has a single root vertex labelled i. The vertices V of T may be partitioned by depth: V = n=0 V n, where the elements of V n are the nth generation individuals of the underlying Galton Watson process. By (H1), all of the generations V n are finite, and by convention V 0 is a singleton containing the root vertex x 0. The (directed) edges of T connect those ordered pairs (v, w) of vertices such that w is an offspring of v. If there is a directed path starting at a vertex v and ending at a vertex w then w is called a descendant of v; the subtree of T whose vertices are the descendants of v

3 Extension of Hawkes theorem 43 is denoted by T(v). An infinite (connected) path in T starting in V 1 and passing through exactly one vertex in each generation V n,n 1, is called an end of T, or an infinite descent line. The space of ends of T (the limit set of G) is denoted by ; it is endowed with the metric d defined by (1) above. For each vertex v, let (v) be the set of all ends that pass through v, and let τ(v) be the type of v. For each end ξ = ξ 1 ξ 2, the sequence of types τ(ξ n ) will be denoted by (ξ), and will be called the pedigree of the descent line. Define ψ : by ψ(ω) = log q(ω 1,ω 2 ). For any Borel probability measure µ on, and any µ-integrable function f :, we will write E µ f = fdµ. Theorem 1. Let be the limit set of a multi-type Galton Watson process G satisfying hypotheses (H1) (H2), and let µ be an ergodic, shift-invariant probability measure on with entropy h(µ). Ifh(µ) + E µ ψ<0then the set ( ) µ is almost surely empty. If h(µ) + E µ ψ 0 then almost surely on the event of non-extinction the set 1 ( µ ) has Hausdorff dimension δ H ( 1 ( µ )) = h(µ) + E µψ log 2. (3) The next result concerns the Hausdorff dimensions of sets 1 (U), where U is a set defined by weaker constraints on ergodic averages. Let σ : be the forward shift mapping, let f : k be a continuous vector-valued function, and let J be a nonempty compact subset of k. Call a σ -invariant probability measure µ admissible if it is ergodic and if µ( (ij)) = 0 for every pair ij of types for which q(i,j) = 0 (note that no such pair can appear in sequence in the pedigree of an infinite descent line). Define S n f = f + f σ + f σ 2 + +f σ n 1, A n f = S n f/n, { } U(f,J) = ω : lim distance(a nf(ω),j)=0, and n M(f, J ) = { admissible µ : E µ f J }. Theorem 2. Let f : k be a function such that, for some finite integer m, f(ω) is a function only of the first m entries of the argument ω. Let J be a nonempty compact subset of k such that, for every x J, there exists an admissible probability measure µ on such that E µ f = x. Then the functional µ h(µ) + E µ ψ attains its maximum on M(f, J ),

4 44 S. P. Lalley, T. Sellke max (h(µ) + E µψ)<0 1 (U(f, J )) = a.s., and (4) µ M(f,J ) ( ) h(µ) + δ H ( 1 Eµ ψ (U(f, J ))) = max (5) µ M(f,J ) log 2 almost surely on the event of survival. + Remarks (A) In fact, Theorem 2 holds more generally for functions f : k that are Hölder continuous (with respect to the metric d ). The proof, however, is considerably more technical than the proof in the case where f depends only on finitely many coordinates. (B) Hawkes theorem is a special case of Theorem 2: Let L, the space of types, be a singleton; let f 1; and let J = {1}. Since L is a singleton, there is only one invariant probability measure on = L, and it has entropy zero. Moreover, ψ log α, where α is the mean offspring number. Thus, (5) reduces to Hawkes formula. (C) Suppose that the space L of types has cardinality greater than 1. Let µ be the unique ergodic, σ -invariant probability measure on that maximizes h(µ) + E µ ψ (that there is a unique maximizer follows, e.g., from Theorem 1.22 of [2]). Now let f 1 and J = {1}; then U(f,J) =, and M(f, J ) is the set of all ergodic, σ -invariant probability measures on. Consequently, Theorems 1 and 2 imply that, if h(µ ) + E µ ψ>0, then almost surely on the event of nonextinction, δ H ( ) = δ H ( 1 ( µ )). (6) Thus, the set of infinite descent lines ζ such that (ζ) is generic for the maximizing measure µ has Hausdorff dimension as large as the entire set of infinite descent lines. (D) Now let f : be any continuous function, and let J be a closed interval of not containing E µ f. Then by Theorem 2 and the fact that h(µ) + E µ ψ is uniquely maximized by µ, δ H ( 1 (U(f, J ))) < δ( ). (7) Thus, the set of infinite descent lines ζ such that (ζ) is not nearly generic for µ has Hausdorff dimension strictly less than the entire set of infinite descent lines. Together with remark (B), this suggests that in an appropriate sense most infinite descent lines are almost generic for µ. The remainder of the paper will be devoted to the proofs of Theorems 1 and 2.

5 Extension of Hawkes theorem Proof of Theorem 1: upper bound In this section we shall prove that δ H ( 1 ( µ )) (h(µ) + E µ ψ)/ log 2 and (8) h(µ) + E µ ψ<0 1 ( µ ) = (9) almost surely. As is often the case in Hausdorff dimension calculations, this is the easy half of the argument: we need only find an efficient sequence of open coverings of the target set 1 ( µ ). Lemma 1. Let µ be any ergodic, invariant probability measure on = L. Then for every ε>0there exist sets Ɣ m = Ɣ m (µ) L m such that lim sup m 1 m x Ɣ m lim m m 1 log Ɣ m =h(µ), (10) m 1 j=0 and such that for every x = x 1 x 2, ψ(σ j x) ψdµ =0, (11) x µ x 1 x 2 x m Ɣ m infinitely often. (12) Note. In fact, there exist sets Ɣ m such that (11) and (10) hold, and for which (12) holds with infinitely often replaced by eventually. This stronger statement will not be needed in the subsequent arguments. Proof. This is a routine consequence of the Shannon-MacMillan theorem and the ergodic theorem (see, e.g., [11]). These two theorems, together with the ergodicity of µ, imply that there exist sets Ɣ m L m such that lim µ( x Ɣ m m (x)) = 1, (13) lim max m 1 log µ( (x)) h(µ) =0, (14) m x Ɣ m and such that (11) holds. Observe that, by (14), for each ε>0, if m is sufficiently large then for every x Ɣ m, and so exp { m(1 + ε)h(µ)} µ( (x)) exp { m(1 ε)h(µ)}, Ɣ m exp { m(1 + ε)h(µ)} µ( x Ɣm (x)) Ɣ m exp { m(1 ε)h(µ)}, which, in view of (13), implies (10). To see that (12) must hold, take a subsequence m k of such that (1 µ( x Ɣmk (x))) <. k=1

6 46 S. P. Lalley, T. Sellke By the Borel-Cantelli Lemma, for µ-almost every sequence x = x 1 x 2, the prefix x 1 x 2 x mk is an element of Ɣ mk for all sufficiently large k. For any nth generation individual x V n, define the pedigree (x) of x to be the length-n sequence w = w 1 w 2 w n recording the types of x and its ancestors; thus, w k is the type of its ancestor in generation k. For any word w L n, define V(w) = 1 ( (w)) to be the set of all nth generation individuals with pedigree w. Observe that 1 ( (w)) = (x), (15) x V(w) and that each set (x) in this union has diameter (in the metric d = d defined by (1)) less than 2 n. Observe also that for any word w = w 1 w 2 w n, n 1 n E V(w) =q(i,w 1 ) q(w j,w j+1 )=q(i,w 1 )exp ψ(σ j 1 w). j=1 (16) Fix ε>0, and let Ɣ m = Ɣ m (µ) be as in the statement of Lemma 1. Then by assertion (12), for each m = 1, 2,..., 1 ( µ ) n=m x Ɣ n ξ V(x) j=1 (ξ) (17) is a covering of 1 ( µ ) by open sets of diameter less than 2 m. Denote this covering by C m. Each C m includes infinitely many open sets (ξ), but by (16), the expected number of these with diameter 2 n is n q(i,x 1 )exp ψ(σ j 1 x), (18) x Ɣ n j=1 which by assertions (10) (11) of Lemma 1 is, for any ε>0, no larger than { ( )} exp n h(µ) + ψdµ + ε (19) provided n is sufficiently large. Proof of (9). Suppose that 2ε = h(µ) + E µ ψ<0. Then by (19), for m sufficiently large, the expected number of sets T(x) included in the covering C m is smaller than e nε. n=m Since this converges to zero exponentially fast as m, the Borel-Cantelli Lemma imples that, with probability one, for all sufficiently large m, the covering C m is empty. It therefore follows that with probability one, 1 ( µ ) =.

7 Extension of Hawkes theorem 47 Proof of (8). Recall that the Hausdorff dimension of a set F is defined to be inf {α : H α (F ) < }, where H α denotes outer α-dimensional Hausdorff measure. Recall also that H α (F ) is the limit as ρ 0 of inf( G G diameter(g)α ), where the infimum is taken over all coverings G of F by open sets G of diameter less than ρ. By (17) and (19), for any α > ((h(µ) + E µ ψ)/ log 2, E diameter( (ξ)) α κ(α) <, (ξ) C m for some κ(α) < independent of m. Consequently, the outer α-dimensional Hausdorff measure of 1 ( µ ) is almost surely finite, and so the advertised inequality (8) must hold almost surely. 3. Proof of Theorem 1: lower bound 3.1. Basic strategy In this section, we prove the more difficult inequality δ H ( 1 ( µ )) (h(µ) + E µ ψ)/ log 2 (20) almost surely on the event of nonextinction. It suffices to prove this for those ergodic, shift-invariant probability measures µ on such that h(µ) + E µ ψ>0 (21) because for those µ not satisfying this condition the inequality (20) is trivial. Thus, for the remainder of this section we assume that (21) holds. Recall that the root of the tree T is of type i. By hypothesis (H2), on the event that T is infinite, it almost surely has infinitely many vertices of type i. For each vertex x T of type i, the subtree T(x) containing those vertices descending from x has the same distribution as the entire tree T, since the root vertex of T has type i. It follows by a routine argument that if for some constant γ > 0 the Hausdorff dimension of 1 ( µ ) exceeds γ with positive probability, then almost surely on the event that T is infinite, the Hausdorff dimension of 1 ( µ ) exceeds γ ; thus, δ H ( 1 ( µ ) is almost surely constant on the event of nonextinction. Consequently, to establish the inequality (20) it suffices to show that with positive probability there is a subtree T of T whose set of ends satisfies 3.2. Background ( ) µ and (22) δ H ( ) (h(µ) + E µ ψ)/ log 2. (23) First we collect some needed results concerning the Hausdorff dimension of nearly regular trees and pruned Galton Watson trees.

8 48 S. P. Lalley, T. Sellke Nearly regular trees Let T be any infinite rooted tree, with vertex set V partitioned by depth (distance from the root vertex): V = n=0 V n. Let N n and M n be increasing sequences of positive integers. Say that T is {M n }-regular relative to {N n } if, for every n 1, every vertex x at depth N n has exactly M n+1 descendant vertices at depth N n+1.if T is {M n }-regular relative to {N n } for a constant sequence M n = D, say that T is D-regular relative to {N n }. Lemma 2. Let T be an infinite rooted tree with space of ends T. If there is an increasing sequence of integers N n satisfying lim n N n+1 /N n = 1 such that, for some sequence {M n } of positive integers, T is {M n }-regular with respect to {N n }, then the Hausdorff dimension of T (relative to the usual metric) is δ H ( T ) = lim inf n log V Nn N n log 2 = lim inf n log n j=1 M n N n log 2. Proof. Easy exercise Pruning a Galton Watson tree Let T = T Z be the genealogical tree of an ordinary Galton Watson process Z with mean offspring number α>1 and survival probability ρ>0. Let the vertex set V of T be partitioned by depth (generation): V = n=0 V n. Pruning Procedure. Let N m and M m be sequences of positive integers. The pruning procedure consists of an infinite sequence of cuts, each depending on the outcome of the earlier cuts. The set of vertices surviving cut n will be denoted by G n, and G 0 = V. For each n 0, the set G n+1 is defined to be the set obtained from G n by removing, for each m, all vertices x G n at depth N m that have fewer than M m descendant vertices in G n at depth N m+1. Note that if the root vertex survives all the cuts, then T has a subtree that is {M m }-regular relative to {N m }, and whose vertices are all elements of n 0 G n. Lemma 3. Suppose that there exist positive probabilities p m such that for each m 0, P { V Nm+1 =k } k P ζ m+1,j M m p m, (24) k=0 j=1 where ζ m,j are mutually independent Bernoulli-p m random variables that are independent of the first N m generations of the Galton Watson process Z. Then the probability that the root vertex survives all the cuts of the pruning procedure is at least p 0. Proof. We will show by induction on n that, for each vertex x V Nm, the conditional probability that x G n, given the first N m generationsof Z, is at least p m. Since

9 Extension of Hawkes theorem 49 the sets G n are decreasing in n, it will follow that x n=1 G nwith conditional probability at least p m. This will prove the lemma. For any vertex x V Nm, the event that x survives the initial cut has probability 1. For each x m V Nm and each n 1, define Yx n = 1 ifx G n, Yx n = 0 ifx G n. Then for each pair m, n the random variables { Yx n : x V } N m are conditionally independent and identically distributed, given the first N m generations of Z. Moreover, by the induction hypothesis, for each vertex x V Nm the conditional probability that Yx n = 1 is at least p m. A vertex x V Nm survives cut n + 1 if and only if Yy n M m, y V Nm+1 (x) where the sum is over the set V Nm+1 (x) of vertices y V Nm+1 that are descendants of x. By hypothesis (24), this event has conditional probability at least p m,given the first N m generations of Z. Verification of the hypothesis (24) in paragraph 3.5 below will be based on the following simple lemma. Recall that ρ>0 is the survival probability of the Galton Watson process Z. Lemma 4. Fix 1 <β<αand 0 <p 1. Suppose that to the vertices v V are attached random variables ξ v so that {ξ v : v V n } are, for each integer n 0, conditionally independent Bernoulli-p, given V n. Define events Then F (n, p, β) = ξ v >β n. (25) v V n lim P (F (n, p, β)) = ρ. (26) n Proof. On the event of survival, V n /β n almost surely as n. Conditional on the event V n =k, the random variable v V n ξ v has the binomial distribution with parameters k and p. Therefore, the lemma follows by an easy application of the Chebyshev inequality. Remark. The probabilities P (F (n, p, β)) are nondecreasing in p.

10 50 S. P. Lalley, T. Sellke 3.3. µ-generic subtrees Recall that L is the set of types of the multi-type Galton Watson process G in Theorems 1 2, T is the genealogical tree of G, and is the space of ends (equivalently, infinite descent lines) of T. Let ε m > 0 be a sequence of real numbers decreasing to 0 as m, and let n m = n(m) m be an increasing sequence of positive integers. Define F m to be the set of all words w L n(m) such that (a) w ends with the letter i ; (b) w begins with a letter w 1 for which q(i,w 1 )>0; (c) no two-letter word u = ij for which q(i,j) = 0 appears in w; and (d) for each word u m j=1 Lj, the relative frequency of u in w differs from µ( (u)) by less than ε m. Observe that if u = ij is a two-letter word for which q(i,j) = 0 then µ( (u)) = 0, because E µ ψ >. Now let k m = k(m) be any sequence of positive integers. Define to be the subset including those ends ξ for which the pedigree (ξ) consists of k 1 words in F 1, followed by k 2 words of F 2, etc. The following lemma is an obvious consequence of the assumption that lim m ε m = 0, together with the definitions of the sets F m and µ. Lemma 5. ( ) µ. The remainder of the argument entails showing that the as yet unspecified sequences k m and n m can be chosen so that with positive probability, T has a subtree T whose set of ends is contained in and satisfies (23). For this, the following lemma is crucial. Lemma 6. For any sequence ε m > 0 such that lim m ε m = 0, if n m sufficiently rapidly then 1 lim inf log n(m) 1 q(w i,w i+1 ) h(µ) + E µ ψ>0. (27) m n m w F m i=1 Proof. By definition of F m, for every w F m and every two-letter word u, the relative frequency of u in w differs from µ( (u)) by less than ε m. Consequently, there exist constants δ m > 0, depending only on ε m and the mean offspring numbers q ij, such that lim m δ m = 0 and for every w F m, n(m) 1 i=1 q(w i,w i+1 ) exp { n m (E µ ψ δ m ) }. The Shannon-MacMillan(-Breiman) Theorem (see, e.g., [11]) and the Ergodic Theorem imply that for any ε m > 0, if n m is sufficiently large then there is a subset Fm of F m such that µ( (w)) µ( (i ))/2 w F m

11 Extension of Hawkes theorem 51 and such that for all w F m, µ( (w)) exp { n m (h(µ) + δ m )}. It follows that F m exp {n m (h(µ) δ m )} µ( (i )/2. This, together with the inequality of the preceding paragraph, implies the inequality (27), since lim m δ m = 0. Remark. The sequence {n m } m 1 may always be chosen so as to be strictly increasing. Moreover, it may be replaced by the sequence {n m+k } m 1, and so it may be arranged that the infimum of the sequence on the left side of (27) may be made arbitrarily close to h(µ) + E µ ψ. Since by hypothesis this is positive, we may choose {n m } m 1 so that, for every m 1, n(m) 1 α m = q(w i,w i+1 )>1, (28) where w 0 = i. w F m i= Embedded Galton Watson processes Recall that the vertex set V of T is partitioned by depth: V = n=0 V n; and that V 0 consists of a single vertex of type i. For each m 1 define subsets Uj m V jn(m) inductively as follows: (a) U0 m = V 0; and (b) for each vertex x V (j+1)n(m), x is included in Uj+1 m if and only if (i) it is a descendant (in T) of a vertex in Uj m; and (ii) the last n m letters of the pedigree (x) form a word in F m. Lemma 7. For each m, the sequence of random variables ( U m j ) j 0 is an ordinary Galton Watson process with mean offspring number α m defined by (28). Proof. Every vertex x Uj m has a pedigree (x) ending in the letter i. Consequently, for each such x, the set T(x) of descendants of x in T constitutes a realization of the multitype Galton Watson process G. Moreover, these realizations are conditionally independent, given the first jn m generations of G, since G is a (multi-type) Galton Watson process. Thus, ( Uj m ) j 0 is an ordinary Galton Watson process. That the mean offspring number is given by the sum in (28) is apparent from the construction. For each m, the mean offspring number α m of the Galton Watson process Uj m is, by (28), larger than 1; hence, Uj m is supercritical. Define { } ρ m = ρ(m) = P lim j m = j (29) to be its survival probability.

12 52 S. P. Lalley, T. Sellke 3.5. Existence of nearly regular subtrees Using the embedded Galton Watson processes defined in paragraph 3.4, we will show that integers k m may be chosen so that if is defined as in paragraph 3.3, then with positive probability T has a subtree T whose set of ends is contained in and satisfies (23). This will complete the proof of (20) and, hence, that of Theorem 1. Define p m = p(m) = min(ρ m,ρ m+1 )/2. (30) By Lemma 6, there exist integers β m so that for each m 1, β m <α m and (31) log β m lim = h(µ) + E µ ψ. (32) m n m Now suppose, as in Lemma 4, that to each vertex v j Uj m is attached a random variable Y v, so that for each generation j the random variables Y v, where v Uj m, are conditionally independent given Uj m, each with the Bernoulli-p m distribution. By Lemma 4, there exist integers r m = r(m) such that for each m, where P(F m (r m,p m,β m ))>ρ m /2 (33) F m (r, p m,β m )= Y v >βm r. v U m j We may assume that the sequence r m is increasing. Finally, let k m = s m r m be an integer multiple of r m, with s m chosen so that r m+1 n m+1 lim = 0. (34) m k m n m Lemma 8. If the sequences r m,s m, and k m = s m r m satisfy (33) and (34), and if is defined as in paragraph 3.3, then with positive probability T has a subtree T whose set of ends is contained in and has Hausdorff dimension at least h(µ) + E µ ψ. Proof. For each positive integer j let m = m(j) be the unique positive integer such that m 1 m s i <j s i, and define i=1 i=1 M j = β r m m and ( ) N j = m 1 k i n i + j m 1 s i r m n m. i=1 i=1

13 Extension of Hawkes theorem 53 Relations (34) and (32) imply that N j+1 lim = 1 j N j and (35) log j i=1 lim M i j N j log 2 = h(µ) + E µψ log 2. (36) Therefore, by Lemma 2, it suffices to prove that with positive probability T contains a subtree T whose set of ends is contained in and which is { } M j -regular with respect to { } N j. Let x be a vertex of T at depth N j. Partition the descendants y of x at depth N j+1 into two classes, good and bad : say that y is good if the last N j+1 N j letters of its pedigree are the concatenation of r m words in F m (if N j+1 N j = r m n m )orthe concatenation of r m+1 words in F m+1 (if N j+1 N j = r m+1 n m+1 ); otherwise, say that y is bad. Define Y x to be 1 if x has at least M m good descendants at depth N j+1, and 0 otherwise. Then by (33), the conditional probability that Y x = 1, given the first N j generations of the Galton Watson process G, is at least p m. Consequently, by (33) and Lemma 3, there is positive probability that the root vertex survives all cuts of the pruning procedure described in paragraph On this event, the pruned tree contains a subtree that is { } { } M j -regular with respect to Nj. By construction, the limit set of the pruned tree is contained in. 4. Proof of Theorem 2 Observe first that, by Theorem 1, δ H ( 1 (U(f, J )) is, on the event of survival, at least as large as the maximum over M(f, J ) of (h(µ) + E µ ψ)/ log 2, because, for each µ M(f, J ), the set µ is contained in U(f,J). Thus, to prove equation (5), it suffices to prove the inequality ( ) h(µ) + δ H ( 1 Eµ ψ (U(f, J ))) max. (37) µ M(f,J ) log 2 + Without loss of generality, we may assume that the function f : k is a function only of the first two entries of its argument. This is because, for any integer k 1, the type space L of the underlying multi-type Galton Watson process G may be enlarged to L m, with each individual of G being re-labelled in accordance with its type and the types of its m 1 most recent ancestors. Moreover, we may assume that the components of the vector f are the indicator functions f ij of those pairs ij of types for which q(i,j) > 0. (If f is not originally of this form, it can be transformed by adjoining indicator functions, replacing components by linear combinations of components, and deleting redundant components. None of these transformations affects the truth of the statements in Theorem 2.) Say that a σ -invariant probability measure µ on is Markov if, under µ, the coordinate process is a (1-step) Markov chain, equivalently, if for every finite sequence x 1 x 2 x m such that µ( (x 1 x 2 x m )>0, µ( (x 1 x 2 x m )) µ( (x 1 x 2 x m 1 )) = µ( (x m 1x m )). (38) µ( (x m 1 ))

14 54 S. P. Lalley, T. Sellke Lemma 9. If, for some w k, there exists an admissible, σ -invariant probability measure ν on such that E ν f = w, then there exists an admissible Markov measure µ = µ w such that E µ f = w, and h(µ) h(ν). (39) Proof. For every pair ij L 2 define π(ij) = ν( (ij)) and π(i) = ν( (i)). Then there is a unique Markov measure µ on with the same marginals π(ij) (the transition probabilities are p(i, j) = π(ij)/π(i) for those pairs ij such that π(ij) > 0, and 0 for the rest). Since the components of f are just the indicators of the different allowable pairs ij of types, E µ f = E ν f. Since ν is admissible, so is µ. The inequality (39) is a routine consequence of the Jensen inequality for convex functions. Letting ϕ(t) = tlog t,wehave 1 ( ) µ( (x1 x 2 x n )) h(ν) = lim µ( (x 1 x 2 x n 2 ))ϕ n 2 µ( (x x 1 x 1 x 2 x n 2 )) 2 x n 1 lim ϕ µ( (x 1 x 2 x n )) n 2 x n 1 x n x 1 x 2 x n 2 1 = lim ϕ(µ( (x n 1 x n ))) n 2 x n 1 x n = h(µ). One of the advantages of dealing with Markov measures is that the entropy functional h(µ) varies continuously with the transition probabilities p ij. Hence, it follows from Lemma 9 that, for any nonempty compact set J { E µ f : µ admissible }, the functional µ E µ f + h(µ) attains its maximum in M(f, J ), and does so at an admissible Markov measure. Lemma 10. For each admissible Markov measure µ = µ w and each ε>0there exists δ>0such that for all sufficiently large n, { x L n : A n f(x) w δ } exp {nh(µ) + nε}. (40) Proof. Consider the set N(µ) = N δ (µ) of all Markov measures µ for which the two-step marginal probabilities µ ( (ij)) are all within δ of the corresponding marginal probabilities µ( (ij)) for µ. For any µ N(µ), the marginal probabilities µ ( (ij)) are bounded above by ν(i,j), where ν(i,j) = µ( (ij)) + δ. Define transition probabilities p (i, j) by p (i, j) = ν(i, j)/ν(i), ν(i) = ν(i,j). j

15 Extension of Hawkes theorem 55 Let µ be the Markov measure for which the associated Markov chain has transition probabilities p (i, j). Observe that, since ν(i,j) > 0, the Markov measure µ is mixing. If δ>0 is small then the transition probabilities p (i, j), and consequently also the stationary probabilities π (i), are close to those of the Markov chain asociated with the Markov measure µ. It follows that if δ>0 is small then the differences π (i) ν(i) and h(µ ) h(µ) are small. For any sequence x L n such that A n f(x) w <δ, the empirical distribution of adjacent pairs ij in x must coincide with the two-step marginal probabilities µ ( (ij)) for some µ N(µ). Consequently, by the definitions of the quantities ν and p (i, j), for each such sequence x = x 1 x 2 x n, n 1 m=1p (x m,x m+1 ) i,j p (i, j) nν(i,j) e nε/4 p (i, j) nπ (i)p (i,j) i,j exp { nh(µ ) nε/3} exp { nh(µ) nε/2}, provided δ>0 is sufficiently small and n is large. Since n 1 π (x 1 ) p (x m,x m+1 )=1, x L n m=1 it follows that the number of sequences x L n such that A n f(x) w <δcannot exceed exp {nh(µ) + nε} for large n. Proof of Inequalities (4) and (37). Let J k be a compact set such that for every w J there exists an admissible probability measure µ for which E µ f = w. By Lemma 9, the measure µ w that maximizes entropy subject to the constraint E µ f = w is a Markov measure. By Lemma 10, for each ε>0and each w J there exists δ = δ w > 0 such that for all large n, inequality (40) holds, and so that, for each x = x 1 x 2 x n L n such that A n f(x) w <δ i, n 1 { q(i,x 1 ) q(x j,x j+1 ) exp n j=1 } ψdµ w + nε. (41) Since J is compact, there exists a finite subset {w 1,w 2,...,w r } such that J is covered by the balls B i of radii δ wi centered at the points w i. Hence, U(f,J) r i=1 U(f,B i). As in section 2, for any x L n let V(x) = 1 ( (x)) be the set of all nth generation vertices of T with pedigree x. Then for each m 1, U(f,B i ) n=m x (i,n) ξ V(x) (ξ)

16 56 S. P. Lalley, T. Sellke where (i, n) = { x L n } : A n f(x) w i <δ wi. This is a covering of U(f,B i ) by sets of diameter no larger than 2 m. By (16), (41), and (40), the expected number of sets of diameter 2 n in this covering is no larger than { } exp n(h(µ wi ) + ψdµ wi + 2ε). The proofs of (4) and (37) may now be completed by the same arguments as in section 2. References 1. Athreya, K., and Ney, P.: Branching Processes. Springer-Verlag (1972) 2. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lecture Notes in Math. 470, (1975) 3. Evans, S.: Polar and nonpolar sets for a tree indexed process. Annals of Probability 20(2), (1992) 4. Furstenberg, H.: Intersections of Cantor sets and transversality of semigroups. Problems in analysis (Bochner Symp.) (1970) 5. Hawkes, J.: Trees generated by a simple branching process. J. London Math. Soc.(2) 24, (1981) 6. Lalley, S., Hueter, I.: Anisotropic branching random walks on homogeneous trees. To appear, Probability & Related Fields (MS1247) (1998) 7. Lalley, S., Sellke, T.: Limit set of a weakly supercritical contact process on a homogeneous tree. Annals of Probability 26, (1998) 8. Lyons, R.: Random walks and percolation on trees. Annals of Probability 18, (1990) 9. Pemantle, R., Peres, Y.: Galton-Watson trees with the same mean have the same polar sets. Annals of Probability 23(3), (1995) 10. Pesin, Y.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications Universtiy of Chicago Press, Chicago IL (1997) 11. Petersen, K.: Ergodic Theory. Cambridge Univ. Press, Cambridge UK (1983) 12. Walters, P.: An Introduction to Ergodic Theory. Springer-Verlag, NY (1982)