IMPLICIT RENEWAL THEORY AND POWER TAILS ON TREES PREDRAG R. JELENKOVIĆ, Columbia University

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1 Applied Probability Trust 22 June 212 IMPLICIT RWAL THORY AD POWR TAILS O TRS PRDRAG R. JLKOVIĆ, Columbia University MARIAA OLVRA-CRAVIOTO, Columbia University Abstract We extend Goldie s 1991 Implicit Renewal Theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power tail asymptotics of the distributions of the solutions R to R = D C ir i + Q, R = D C ir i Q, and similar recursions, where Q,, C 1, C 2,... is a nonnegative random vector with {, 1, 2, 3,... } { }, and {R i} i are iid copies of R, independent of Q,, C 1, C 2,... ; here denotes the maximum operator. Keywords: Implicit renewal theory; weighted branching processes; multiplicative cascades; stochastic recursions; power laws; large deviations; stochastic fixed point equations 2 Mathematics Subject Classification: Primary 6H25 Secondary 6J8;6F1;6K5 1. Introduction This paper is motivated by the study of the nonhomogeneous linear recursion R D = C i R i + Q, 1 where Q,, C 1, C 2,... is a nonnegative random vector with { }, = {, 1, 2, 3,... }, P Q > >, and {R i } i is a sequence of iid random variables, independent of Q,, C 1, C 2,..., having the same distribution as R. This recursion appeared recently in the stochastic analysis of Google s PageRank algorithm, see [27, 19 and the references therein for the latest work in the area. These types of weighted recursions, also studied in the literature on weighted branching processes [25 and branching random walks [8, are found in the probabilistic analysis of other algorithms as well [26, 24, e.g., Quicksort algorithm [13. In order to study the preceding recursion in its full generality we extend the implicit renewal theory of Goldie [14 to cover recursions on trees. The extension of Goldie s theorem is presented in Theorem 3.1 of Section 3. One of the observations that allows Postal address: Department of lectrical ngineering, Columbia University, ew York, Y 127 Supported by the SF, grant no. CMMI Postal address: Department of Industrial ngineering and Operations Research, Columbia University, ew York, Y 127 Supported by the SF, grant no. CMMI

2 2 P.R. Jelenković and M. Olvera-Cravioto this extension is that an appropriately constructed measure on a weighted branching tree is a renewal measure, see Lemma 3.1 and equation 9. In the remainder of the paper we apply the newly developed framework to analyze a number of linear and non-linear stochastic recursions on trees, starting with 1. ote that the majority of the work in the rest of the paper goes into the application of the main theorem to specific problems. In this regard, in Section 4, we first construct an explicit solution 17 to 1 on a weighted branching tree and then provide sufficient conditions for the finiteness of moments and the uniqueness of this solution in Lemmas 4.4 and 4.5, respectively. Furthermore, it is worth noting that our moment estimates are explicit, see Lemma 4.3, which may be of independent interest. Then, the main result, which characterizes the power-tail behavior of R is presented in Theorem 4.1. In addition, for integer power exponent α {1, 2, 3,... } the asymptotic tail behavior can be explicitly computed as stated in Corollary 4.1. Furthermore, for non integer α, Lemma 4.1 yields an explicit bound on the tail behavior of R. Related work in the literature of weighted branching processes WBPs for the case when = and Q, {C i } are nonnegative deterministic constants can be found in [25 see Theorem 5, and more recently, for real valued constants, in [5. However, these deterministic assumptions fall outside of the scope of this paper; for more details see the remarks after Theorem 4.1 in Section 4.2. ext, we show how our technique [ can be applied to study the tail asymptotics of the solution to the critical, C i = 1, homogeneous linear equation R D = C i R i, 2 where, C 1, C 2,... is a nonnegative random vector with { } and {R i } i is a sequence of iid random variables independent of, C 1, C 2,... having the same distribution as R. This type of recursion has been studied to a great extent under a variety of names, including branching random walks and multiplicative cascades. Our work is more closely related to the results of [23 and [17, where the conditions for power-tail asymptotics of the distribution of R with power exponent α > 1 were derived. In Theorem 4.2 of Section 4.2 we provide an alternative derivation of Theorem 2.2 in [23 and Proposition 7 in [17. Furthermore, we note that our method yields a more explicit characterization of the power-tail proportionality constant, see Corollary 4.2. For the full description of the set of solutions to 2 see the very recent work in [3. For additional references on weighted branching processes and multiplicative cascades see [2, 23, 22, 28, 24 and the references therein. For earlier historical references see [2, 16, 12. As an additional illustration of the newly developed framework, in Section 5 we study the recursion R = D C i R i Q, 3 where Q,, C 1, C 2,... is a nonnegative random vector with { }, P Q > > and {R i } i is a sequence of iid random variables independent of Q,, C 1, C 2,... having the same distribution as R. We characterize the tail behavior of P R > x in Theorem 5.1. Similarly to the homogeneous linear case, this recursion

3 Implicit Renewal Theory on Trees 3 was previously studied in [6 under the assumption that Q, =, and the {C i } are real valued deterministic constants. The more closely related case of Q and {C i } being random was studied earlier in [18. Furthermore, these max-type stochastic recursions appear in a wide variety of applications, ranging from the average case analysis of algorithms to statistical physics; see [1 for a recent survey. We conclude the paper with a brief discussion of other non-linear recursions that could be studied using the developed techniques, including the solution to R = D C i R i + Q. The majority of the proofs are postponed to Section Model description First we construct a random tree T. We use the notation to denote the root node of T, and A n, n, to denote the set of all individuals in the nth generation of T, A = { }. Let Z n be the number of individuals in the nth generation, that is, Z n = A n, where denotes the cardinality of a set; in particular, Z = 1. ext, let + = {1, 2, 3,... } be the set of positive integers and let U = k= + k be the set of all finite sequences i = i 1, i 2,..., i n, where by convention + = { } contains the null sequence. To ease the exposition, for a sequence i = i 1, i 2,..., i k U we write i n = i 1, i 2,..., i n, provided k n, and i = to denote the index truncation at level n, n. Also, for i A 1 we simply use the notation i = i 1, that is, without the parenthesis. Similarly, for i = i 1,..., i n we will use i, j = i 1,..., i n, j to denote the index concatenation operation, if i =, then i, j = j. We iteratively construct the tree as follows. Let be the number of individuals born to the root node, =, and let { i } i U,i be iid copies of. Define now A 1 = {i + : 1 i }, A n = {i 1, i 2,..., i n U : i 1,..., i n 1 A n 1, 1 i n i1,...,i n 1}. 4 It follows that the number of individuals Z n = A n in the nth generation, n 1, satisfies the branching recursion Z n = i. i A n 1 ow, we construct the weighted branching tree T Q,C as follows. The root node is assigned a vector Q,, C,1, C,2,... = Q,, C 1, C 2,... with { } and P Q > > ; determines the number of nodes in the first generation of T according to 4. ach node in the first generation is then assigned an iid copy Q i, i, C i,1, C i,2,... of the root vector and the { i } are used to define the second generation in T according to 4. In general, for n 2, to each node i A n 1, we assign an iid copy Q i, i, C i,1, C i,2,... of the root vector and construct A n = {i, i n : i A n 1, 1 i n i }; the vectors Q i, i, C i,1, C i,2,..., i A n 1 are chosen independently of all the previously assigned vectors Q j, j, C j,1, C j,2,..., j A k, k n 2. For each node in T Q,C we also define the weight Π i1,...,i n via

4 4 P.R. Jelenković and M. Olvera-Cravioto the recursion Π i1 = C i1, Π i1,...,i n = C i1,...,i nπ i1,...,i n 1, n 2, where Π = 1 is the weight of the root node. ote that the weight Π i1,...,i n is equal to the product of all the weights C along the branch leading to node i 1,..., i n, as depicted in Figure 1. In some places, e.g. in the following section, the value of Q may be of no importance, and thus we will consider a weighted branching tree defined by the smaller vector, C 1, C 2,.... This tree can be obtained form T Q,C by simply disregarding the values for Q and is denoted by T C. Π = 1 Z = 1 Π 1 Π 2 Π 3 Z 1 = 3 Π 1,1 Π 1,2 Π 2,1 Π 3,1 Π 3,2 Π 3,3 Z 2 = 6 Figure 1: Weighted branching tree Studying the tail behavior of the solutions to recursions and fixed point equations embedded in this weighted branching tree is the objective of this paper. 3. Implicit renewal theorem on trees In this section we present an extension of Goldie s Implicit Renewal Theorem [14 to weighted branching trees. The observation that facilitates this generalization is the following lemma which shows that a certain measure on a tree is actually a product measure; a similar measure was used in a different context in [9. Its proof is given in Section 7.1 for completeness. Throughout the paper we use the standard convention α log = for all α >. Lemma 3.1. Let T C be the weighted branching tree defined by the nonnegative vector, C 1, C 2,..., where { }. For any n and i A n, let V i = log Π i. For α > define the measure [ µ n dt = e αt 1V i dt, n = 1, 2,..., i A n and let ηdt = µ 1 dt. Suppose that there exists j 1 with P j, C j > > such that the measure P log C j du, C j >, j is nonarithmetic,

5 Implicit Renewal Theory on Trees 5 [ [ < Cα i log C i < and Cα i = 1. Then, η is a nonarithmetic probability measure on R that places no mass at and has mean u ηdu = Cj α log C j. Furthermore, µ n dt = η n dt, where η n denotes the nth convolution of η with itself. We now present a generalization of Goldie s Implicit Renewal Theorem [14 that will enable the analysis of recursions on weighted branching trees. ote that except for the independence assumption, the random variable R and the vector, C 1, C 2,... are arbitrary, and therefore the applicability of this theorem goes beyond the recursions that we study here. Throughout the paper we use gx fx as x to denote lim x gx/fx = 1. Theorem 3.1. Let, C 1, C 2,... be a nonnegative random vector, where { }. Suppose that there exists j 1 with P j, C j > > such that the measure [ P log C j du, C j >, j is nonarithmetic. Assume further that [ < Cα j log C [ j <, Cα j = 1, Cγ j < for some γ < α, and that R is independent of, C 1, C 2,... with [R β < for any < β < α. If P R > t 1C j R > t tα 1 dt <, 5 then where H < is given by 1 H = [ Cα j log C j P R > t Ht α, t, v α 1 P R > v 1C j R > v dv. Remarks: i As pointed out in [14, the statement of the theorem only has content when R has infinite moment of order α, since otherwise the constant H is zero. ii Similarly as in [14, this theorem can be generalized to incorporate negative weights {C i } at the expense of additional technical complications. However, when the {C i } and R is real-valued, one can use exactly the same proof to derive the asymptotics of P R > t; we omit the statement here since our applications do not require it. iii When the {log C i } are lattice valued, a similar version of the theorem can be derived by using the corresponding Renewal Theorem for lattice random walks. iv It appears, as noted in [14, that some of the early ideas of applying renewal theory to study the power tail asymptotics of autoregressive processes perpetuities is due to [21 and [15. The proof given below follows the corresponding proof in [14. Proof of Theorem 3.1. Let T C be the weighted branching tree defined by the nonnegative vector, C 1, C 2,.... For each i A n and all k n define V i k = log Π i k ;

6 6 P.R. Jelenković and M. Olvera-Cravioto note that Π i k is independent of i k but not of i s for any s k 1. Also note that i n = i since i A n. Let F k, k 1, denote the σ-algebra generated by { i, C i,1, C i,2,... : i A j, j k 1 }, and let F = σ, Ω, Π i 1. Assume also that R is independent of the entire weighted tree, T C. Then, for any t R, we can write P R > e t via a telescoping sum as follows note that all the expectations in 6 are finite by Markov s inequality and 11 P R > e t n 1 = 1Π i k R > e t 1Π i k+1 R > e t 6 k= i k A k i k+1 A k+1 + 1Π i n R > e t i n A n n 1 = i k 1Π i k R > e t 1Π i k C i k,j R > e t k= i k A k + 1Π i n R > e t i n A n n 1 = i k 1R > e t V i k 1C i k,j R > e t V i k k= i k A k F k + 1Π i n R > e t. 7 i n A n ow, define the measures µ n according to Lemma 3.1 and let n ν n dt = µ k dt, gt = e αt P R > e t 1C j R > e t, k= rt = e αt P R > e t and δ n t = e αt 1Π i n R > e t. i n A n Recall that R and i k, C i k,1, C i k,2,... are independent of F k, from where it follows that i k 1R > e t V i k 1C i k,j R > e t V i k F k = e αvi k t g t V i k. Then, for any t R and n, n 1 rt = e αvi k gt V i k + δ n t = g ν n 1 t + δ n t. k= i k A k

7 Implicit Renewal Theory on Trees 7 ext, define the operator ft = t e t u fu du and note that rt = ğ ν n 1 t + δ n t. 8 ow, we will show that one can let n in the preceding identity. To this end, let ηdu = µ 1 du, and note that by Lemma 3.1 η is a nonarithmetic probability measure on R that places no mass at and has mean, µ u ηdu = Cj α log C j >. Moreover, by Lemma 3.1, νdt e αt 1V i k dt = η k dt 9 k= i k A k k= is its renewal measure. Since µ, then f νt < for all t whenever f is directly Riemann integrable. By 5 we know that g L 1, so by Lemma 9.1 from [14, ğ is directly[ Riemann integrable, resulting in ğ νt < for all t. Thus, ğ νt = k= i k A k e αv i k ğt Vi k <, which implies that [ k= i k A k e αv i kğt Vi k exists and, by Fubini s theorem, ğ νt = e αvi kğt V i k k= i k A k = e αvi kğt V i k = lim ğ ν nt. n k= i k A k To see [ that δ n t as n for all fixed t, note that from the assumptions [ < Cα j log C [ j <, Cα j = 1, and Cγ j < for some [ γ < α, there exists < β < α such that Cβ j < 1 by convexity. Then, for such β, t δ n t = e t u e αu Πi n R > e u du e α βt = e α βt i n A n 1 t e βu 1 Π i n R > e u du i n A n min{t,logπi n R} i n A n e βu du eα βt Π i n R β. 1 β i n A n

8 8 P.R. Jelenković and M. Olvera-Cravioto It remains to show that the expectation in 1 converges to zero as n. First note that from the independence of R and T C, Π i n R β = [R β Π i n β, i n A n i n A n where [R β <, for < β < α. For the expectation involving Π i n condition on F n 1 and use the independence of i n 1, C i n 1,1, C i n 1,2,... from F n 1 as follows i n 1 Π i n β = Π i n 1 β C β i n 1,j i n A n i n 1 A n 1 F n 1 i n 1 = Π i n 1 β C β i n 1,j i n 1 A n 1 F n 1 = C β j Π i n 1 β i n 1 A n 1 n = iterating n 1 times. 11 C β j [ Since Cβ j < 1, then the above converges to zero as n. Hence, the preceding arguments allow us to pass n in 8, and obtain rt = ğ νt. ow, by the key renewal theorem for two-sided random walks, see Theorem 4.2 in [7, e t e t v α P R > v dv = rt 1 µ ğu du H, t. Clearly, H since the left-hand side of the preceding equation is positive, and thus, by Lemma 9.3 in [14, P R > t Ht α, t.

9 Implicit Renewal Theory on Trees 9 Finally, H = 1 µ = 1 µ = 1 µ = 1 µ = 1 µ u e t gt e αt e u t gt dt du t e u du dt gt dt P R > e t 1C j R > e t dt v α 1 P R > v 1C j R > v dv. 4. The linear recursion: R = C ir i + Q Motivated by the information ranking problem on the Internet, e.g. Google s PageRank algorithm [19, 27, in this section we apply the implicit renewal theory for trees developed in the previous section to the following linear recursion: R D = C i R i + Q, 12 where Q,, C 1, C 2,... is a nonnegative random vector with { }, P Q > >, and {R i } i is a sequence of iid random variables independent of Q,, C 1, C 2,... having the same distribution as R. ote that the power tail of R in the critical homogeneous case Q was previously studied in [23 and [17. In Section 4.3 we will give an alternative derivation of those results using our method and will provide pointers to the appropriate literature. As for the nonhomogeneous case, the first result we need to establish is the existence and finiteness of a solution to 12. For the purpose of existence we will provide an explicit construction of the solution R to 12 on a tree. ote that such constructed R will be the main object of study of this section. Recall that throughout the paper the convention is to denote the random vector associated to the root node by Q,, C 1, C 2,... Q,, C,1, C,2,.... We now define the process W = Q, W n = i A n Q i Π i, n 1, 13 on the weighted branching tree T Q,C, as constructed in Section 2. Define the process {R n } n according to n R n = W k, n, 14 k=

10 1 P.R. Jelenković and M. Olvera-Cravioto that is, R n is the sum of the weights of all the nodes on the tree up to the nth generation. It is not hard to see that R n satisfies the recursion R n = C,j R n 1 j + Q = C j R n 1 j + Q, n 1, 15 where {R n 1 j } are independent copies of R n 1 corresponding to the tree starting with individual j in the first generation and ending on the nth generation; note that = Q j. Similarly, since the tree structure repeats itself after the first generation, W n satisfies R j W n = i A n Q i Π i C,k Q k,...,in C k,...,ij k=1 k,...,i n A n j=2 = D = C k W n 1,k, 16 k=1 where {W n 1,k } is a sequence of iid random variables independent of, C 1, C 2,... and having the same distribution as W n 1. ext, define the random variable R according to R lim n Rn = n W k, 17 where the limit is properly defined by 14 and monotonicity. Hence, it is easy to verify, by applying monotone convergence in 15, that R must solve R = C,j R j + Q = k= C j R j + Q, where {R j } j are iid, have the same distribution as R, and are independent of Q,, C 1, C 2,.... The derivation provided above implies in particular the existence of a solution in distribution to 12. Moreover, under additional technical conditions, R is the unique solution under iterations as we will define and show in the following section. The constructed R, as defined in 17, is the main object of study in the remainder of this section Moments of W n and R In this section we derive estimates for the moments of W n and R. We start by stating a lemma about the moments of a sum of random variables. The proofs of Lemmas 4.1, 4.2 and 4.3 are given in Section 7.2.

11 Implicit Renewal Theory on Trees 11 Lemma 4.1. For any k { } let {C i } k be a sequence of nonnegative random variables and let {Y i } k be a sequence of nonnegative iid random variables, independent of the {C i }, having the same distribution as Y. For β > 1 set p = β {2, 3, 4,... }, and if k = assume that C iy i < a.s. Then, k β k C i Y i C i Y i β [ k β Y p 1 β/p 1 C i. [ Remark: ote that the preceding lemma does not exclude the case when k [ β k β C iy i = but C k iy i C iy i β <. We now give estimates for the β-moments of W n for β, 1 and β > 1 in Lemmas 4.2[ and 4.3, respectively. Throughout the rest of the paper define ρ β = for any β >, and ρ ρ 1. Cβ i Lemma 4.2. For < β 1 and all n, [W β n [Q β ρ n β. [ β Lemma 4.3. For β > 1 suppose [Q β <, C i <, and ρ ρ β < 1. Then, there exists a constant K β > such that for all n, [W β n K β ρ ρ β n. ow we are ready to establish the finiteness of moments of the solution R given by 17 in Section 4. The proof of this lemma uses well known contraction arguments, but for completeness we provide the details below. Lemma 4.4. Assume that [Q β < for some β > [. In addition, suppose that β either i ρ β < 1 if < β < 1, or ii ρ ρ β < 1 and < if β 1. C i Then, [R γ < for all < γ β, and in particular, R < a.s. Moreover, if β 1, R n L β R, where L β stands for convergence in β 1/β norm. [ Remark: It is interesting to observe that for β > 1 the conditions ρ β < 1 and β < are consistent with Theorem 3.1 in [2, Proposition 4 in [17 C i and Theorem 2.1 in [23, which give the conditions for the finiteness of the β-moment of the solution to the related critical ρ 1 = 1 homogeneous Q equation. Proof. Let η = { ρ β if β < 1 ρ ρ β, if β 1. Then by Lemmas 4.2 and 4.3, [W β n Kη n 18

12 12 P.R. Jelenković and M. Olvera-Cravioto for some K >. Suppose β 1, then, by monotone convergence and Minkowski s inequality, n β n β [R β = lim W k = lim W k n n lim n n k= k= [W β k 1/β β K k= β η k/β <. This implies that R < a.s. When < β 1 use the inequality n k= y k β n k= yβ k for any y i instead of Minkowski s inequality. Furthermore, for any k= < γ β, [ [R γ = R β γ/β [R β γ/β <. L β That [ R n R whenever β 1 follows from noting that [ R n R β = k=n+1 W β k and applying the same arguments used above to obtain the bound [ R n R β Kη n+1 /1 η 1/β β. ext, we show that under some technical conditions, the iteration of recursion 12 results in a process that converges in distribution to R for any initial condition R. To this end, consider a weighted branching tree T Q,C, as defined in Section 2. ow, define where R n 1 is given by 14, R n R n 1 + W n R, n 1, W n R = i A n R,iΠ i, 19 and {R,i } i U are iid copies of an initial value R, independent of the entire weighted tree T Q,C. It follows from 15 and 19 that, for n, Rn+1 = C j R n 1 j +Q+W n+1 R = C j + n C j,...,ik +Q, R n 1 j R,i i A n,j k=2 2 where {R n 1 j } are independent copies of R n 1 corresponding to the tree starting with individual j in the first generation and ending on the nth generation, and A n,j is the set of all nodes in the n + 1th generation that are descendants of individual j in the first generation. It follows that R n+1 = C j Rn,j + Q, where {R n,j } are the expressions inside the parenthesis in 2. Clearly, {R n,j } are iid copies of R n, thus we show that R n is equal in distribution to the process derived by iterating 12 with an initial condition R. The following lemma shows that R n R for any initial condition R satisfying a moment assumption, where denotes convergence in distribution.

13 Implicit Renewal Theory on Trees 13 Lemma [ 4.5. For any initial condition R, if [Q β, [R β < and ρ β = < 1 for some < β 1, then Cβ i R n R, with [R β <. Furthermore, under these assumptions, the distribution of R is the unique solution with finite β-moment to recursion 12. Proof. Since R n R a.s., the result will follow from Slutsky s Theorem see Theorem 25.4, p. 332 in [1 once we show that W n R. To this end, note that W n R, as defined by 19, is the same as W n if we substitute the Q i by the R,i. Then, for every ɛ > we have that P W n R > ɛ ɛ β [W n R β ɛ β ρ n β[r β by Lemma 4.2. Since by assumption the right-hand side converges to zero as n, then R n R. Furthermore, [R β < by Lemma 4.4. Clearly, under the assumptions, the distribution of R represents the unique solution to 12, since any other possible solution with finite β-moment would have to converge to the same limit. Remarks: i ote that when [ < 1 the branching tree is a.s. finite and no conditions on the {C i } are necessary for R < a.s. This corresponds to the second condition in Theorem 1 of [11. ii In view of the same theorem from [11, one could possibly establish the convergence of Rn R < under milder conditions. However, since in this paper we only study the power [ tails of R, the assumptions of Lemma 4.5 are not restrictive. iii ote that if Cα i = 1 with α, 1, then there might [ not be a < β < α for which < 1, e.g., the case of deterministic C i s that was studied in [ Main result Cβ i We now characterize the tail behavior of the distribution of the solution R to the nonhomogeneous equation 12, as defined by 17. Theorem 4.1. Let Q,, C 1, C 2,... be a nonnegative random vector, with { }, P Q > > and R be the solution to 12 given by 17. Suppose that there exists j 1 with P j, C j > > such that the measure P log [ C j du, C j >, j is nonarithmetic, and that for some α >, [Q α <, < Cα i log C i < [ and = 1. In addition, assume Cα i [ [ 1. C α i < 1 and C i <, if α > 1; or, [ 1+ɛ 2. Cα/1+ɛ i < for some < ɛ < 1, if < α 1. Then, P R > t Ht α, t,

14 14 P.R. Jelenković and M. Olvera-Cravioto where H < is given by [ 1 H = [ v P α 1 R > v 1C i R > v dv Cα i log C i [ α C ir i + Q C ir i α = [. α Cα i log C i Remarks: i The nonhomogeneous equation has been previously studied for the special case when Q and the {C i } are deterministic constants. In particular, Theorem 5 of [25 analyzes the solutions to 12 when Q and the {C i } are nonnegative deterministic constants, which, when Cα i = 1, α >, implies that C i 1 for all i and i Cα i log C i, falling outside of the scope of this paper. The solutions to 12 for the case when Q and the C i s are real valued deterministic constants were analyzed in [5. For the very recent work published on arxiv after the first draft of this paper that characterizes all the [ solutions to 12 for Q and {C i } random see [4. ii When α α > 1, the condition C i < is needed to ensure that the tail of R is not dominated by. In particular, if the {C i } are iid and independent of, the condition reduces to [ α < since [C α < is implied by the other conditions; see [ Theorems 4.2 and 5.4 in [19. Furthermore, when < α 1 the condition α [ C α [ i < is redundant since C i Cα i = 1, and [ 1+ɛ the additional condition < is needed. When the {C i } are Cα/1+ɛ i iid and independent of, the latter condition reduces to [ 1+ɛ < given the other assumptions, which is consistent with Theorem 4.2 in [19. iii ote that the second expression for H is more suitable for actually computing it, especially in the case of α being an integer, as will be stated in the forthcoming Corollary 4.1. When α > 1 is not an integer, we can derive an explicit upper bound on H by using Lemma 4.6. k α Regarding the lower bound, the elementary inequality x k i xα i for α 1 and x i, yields H α [Q α [ Cα i log C i >. k α Similarly, for < α < 1, using the corresponding inequality x k i xα i [ for < α 1, x i, we obtain H [Q α / α Cα i log C i. iv Let us also observe that the solution R, given by 17, to equation 12 may be a constant non power law R = r > when P r = Q + r C i = 1. However, similarly as in remark i, such a solution is excluded from the theorem since P r = Q+r C i = 1 implies [ i Cα i log C i, α >. Before proceeding with the proof of Theorem 4.1, we need the following two technical results; their proofs are given in Section 7.3. Lemma 4.6 below will also be used in subsequent sections for other recursions. With some abuse of notation, we will use throughout the paper max 1 i x i to denote sup 1 i<+1 x i in case =.

15 Implicit Renewal Theory on Trees 15 Lemma 4.6. Suppose, C 1, C 2,... is a nonnegative random vector, with { } and let {R i } i be a sequence of iid nonnegative random variables independent of, C 1, C 2,... having the same distribution as R. For α >, suppose that C [ ir i α < a.s. and [R β < for any < β < α. Furthermore, assume 1+ɛ that < for some < ɛ < 1. Then, Cα/1+ɛ i [ 1C i R i > t P max C ir i > t t α 1 dt 1 i [ = 1 α α C i R i α max C ir i <. 1 i Lemma 4.7. Let Q,, C 1, C 2,... be a nonnegative vector with { } and let {R i } be a sequence of iid random variables, independent [ of Q,, C 1, C 2,.... Suppose α that for some α > 1 we have [Q α <, C i <, [R β < for any < β < α, and C ir i < a.s. Then [ α C i R i + Q C i R i α <. Proof of Theorem [ 4.1. By Lemma 4.4, we know that [R β < for any < β < α. To verify that Cγ i < for some γ < α note that if α > 1 we have, by the assumptions of the theorem and Jensen s inequality, [ [ γ [ α γ/α C i C i < C γ i for any 1 γ < α. If < α 1, then for γ = α1 + ɛ/2/1 + ɛ < α we have [ C γ i C α/1+ɛ i 1+ɛ/2 C α/1+ɛ i 1+ɛ 1+ɛ/2 1+ɛ <. The statement of the theorem with the first expression for H will follow from Theorem 3.1 once we prove that condition 5 holds. To this end, define R = C i R i + Q. Then, [ P R > t 1C i R i > t P R > t P max C ir i > t 1 i [ + P max C ir i > t 1C i R i > t 1 i.

16 16 P.R. Jelenković and M. Olvera-Cravioto Since R = D R max 1 i C i R i, the first absolute value disappears. For the second one, note that [ 1C i R i > t P max C ir i > t 1 i [ = 1C i R i > t [ 1 max C ir i > t. 1 i ow it follows that [ P R > t 1C i R i > t P R > t P max C ir i > t 1 i [ + 1C i R i > t P max C ir i > t i ote that the integral corresponding to 21 is finite by Lemma 4.6 if we show that the assumptions of Lemma 4.6 are satisfied when α > 1. ote that in this case we can choose ɛ > such that α/1 + ɛ 1 and use the inequality k C α/1+ɛ i x β i k x i β 22 for β 1, x i, k to obtain 1+ɛ [ α C i <. Therefore, it only remains to show that P R > t P max C ir i > t t α 1 dt < i To see this, note that R = D R and 1R > t 1max 1 i C i R i > t, and thus, by Fubini s theorem, we have P R > t P max C ir i > t t α 1 dt = 1α [ α R α max C ir i. 1 i 1 i If < α 1, we apply 22 to obtain [ α [ R α max C ir i Q α + 1 i α C i R i α max C ir i, 1 i

17 Implicit Renewal Theory on Trees 17 which is finite by Lemma 4.6 and the assumption [Q α <. k α If α > 1, we have x k i xα i, x i, k, implying that we can split the expectation as follows [ α [ R α max C ir i = R α 1 i C i R i α [ α + C i R i α max C ir i, 1 i which can be done since both expressions inside the expectations on the right-hand side are nonnegative. The first expectation is finite by Lemma 4.7 and the second expectation is again finite by Lemma 4.6. Finally, applying Theorem 3.1 gives where H = P R > t Ht α, [ 1 Cα j log C j v P α 1 R > v To obtain the second expression for H note that v α 1 = P R > v 1C j R > v dv [ v α 1 1 C i R i + Q > v 1C i R i > v dv [ 1C jr > v dv. [ = v 1 α 1 C i R i + Q > v 1C i R i > v dv [ CiRi+Q = v α 1 dv CiR i [ = 1 α α C i R i + Q C i R i α, v α 1 dv where 24 is justified by Fubini s Theorem and the integrability of v α 1 1 C i R i + Q > v 1C i R i > v v 1 α 1 C i R i + Q > v 1 max C ir i > v 1 i + v α 1 1C i R i > v 1 max C ir i > v, 1 i 24 25

18 18 P.R. Jelenković and M. Olvera-Cravioto which is a consequence of 23 and Lemma 4.6; and 25 follows from the observation that v α 1 1 C i R i + Q > v and v α 1 1C i R i > v are each almost surely absolutely integrable with respect to v as well. This completes the proof. As indicated earlier, when α 1 is an integer, we can obtain the following explicit expression for H. Corollary 4.1. For integer α 1, and under the same assumptions of Theorem 4.1, the constant H can be explicitly computed as a function of [R k, [C k, [Q k, k α 1. In particular, for α = 1, and for α = 2, [Q 2 + 2[R H = H = [ Q [Q [R = [. 1 C i [Q [, C i log C i C i 2 [ + 2[R 2 j=i+1 C ic j [, C2 i log C i Proof. The proof follows directly from multinomial expansions of the second expression for H in Theorem The homogeneous recursion In this section we briefly describe how the methodology [ developed in the previous sections can be applied to study the critical, C i = 1, homogeneous linear recursion R = D C i R i, 26 where, C 1, C 2,... is a nonnegative random vector with { } and {R i } i is a sequence of iid random variables independent of, C 1, C 2,... having the same distribution as R. This equation has been studied extensively in the literature under various different assumptions; for recent results see [23, 17, 2 and the references therein. Based on the model from Section 4 we can construct a solution to 26 as follows. Consider the process {W n } n defined by 13 with Q i 1. Then, the [ {W n } satisfy in distribution the homogeneous recursion in 16 and, given that C i = 1, we have [W n = 1. Hence, {W n } n is a nonnegative martingale and by the martingale convergence theorem W n R a.s. with [R 1. ext, provided that [ [ C i log C i < and C i log + C i <

19 Implicit Renewal Theory on Trees 19 it can be shown that [R = 1, see Theorem 1.1d in [2 see also Theorem 2 in [23; log + x = maxlog x,. Furthermore, as argued in equation 1.9 of [2, it can easily be shown that this R is a solution to 26. ote that the same construction of the solution R on a branching tree was given in [2 and [23. Since the solutions to 26 are scale invariant, this construction also shows that for any m > there is a solution R with mean m; or equivalently, it is enough to study the solutions with mean 1. Moreover, under additional assumptions it can be shown that this constructed R is the only solution with mean 1, e.g. see [22, 23, 17. However, it is not the objective of this section to study the uniqueness of this solution, rather we focus on studying the tail behavior of any such possible solution since our Theorem 3.1 does not require the[ uniqueness of R. As a side note, we point out that 26 can have solutions if Cβ i = 1 for some < β < 1, as studied in [22, 17. A version of the following theorem, with a possibly less explicit constant, was previously proved in Theorem 2.2 in [23 and Proposition 7 in [17; they also study the lattice case. Regarding the lattice case, as pointed out earlier in the remark after Theorem 3.1, all the results in this paper can be developed for this case as well by using the corresponding renewal theorem. Theorem 4.2. Suppose that there exists j 1 with P j, C j > > such that the measure P log [ C j du, C j >, j is nonarithmetic. Suppose further that for α [ some α > 1, C [ i <, Cα i log+ C i < and C i = [ = 1. Then, equation 26 has a solution R with < [R < such that Cα i where H < is given by 1 H = [ Cα i log C i = [ P R > t Ht α, t, α C ir i C ir i α [. α Cα i log C i [ v P α 1 R > v 1C i R > v dv Furthermore, if P Ñ 2 >, Ñ = 1C i >, then H >. [ Proof. By the assumptions, the function ϕθ is convex, finite, and Cθ j continuous on [1, α, since ϕ1 = ϕα = 1. Furthermore, by standard arguments, it can be shown that both ϕ θ and ϕ θ exist on the open interval 1, α and, in particular, [ ϕ θ = Ci θ log C i 2. Clearly, ϕ θ > provided that [ P C i {, 1}, 1 i < 1. To see that this is indeed the case, note that C i = 1 implies that P C i, 1 i < 1, which combined with the nonarithmetic assumption yields P C i {, 1}, 1 i <

20 2 P.R. Jelenković and M. Olvera-Cravioto 1. Hence, there exists 1 < θ 1 < θ 2 < α such that ϕ θ 1 < and ϕ θ 2 >, implying by the monotonicity of ϕ and monotone convergence that [ [ < ϕ α = Ci α log C i Ci α log + C i < and 27 [ ϕ 1+ = C i log C i <. [ The last expression and the observation C i log + C i < implied [ α by C i < yields, as argued at the beginning of this section, that recursion 26 has a solution with finite positive mean, see Theorem 1.1d and equation 1.9 in [2 see also Theorem 2 in [23. ext, in order to apply Theorem 3.1, we use 27 and [R β < for any < β < α; the latter follows from Theorem 3.1 in [2 and the strict convexity of ϕ argued above see also, Proposition 4 in [17 and Theorem 2.1 in [23. The rest of the proof, except for the H > part, proceeds exactly as that of Theorem 4.1 by setting Q. Regarding the H > statement, the assumption P Ñ 2 > implies that there exist 1 n and 1 i 1 < i 2 < n + 1 such that P = n, C i1 >, C i2 > >, which further yields, for some δ >, P i 2, C i1 > δ, C i2 > δ >. 28 ext, by using the inequality x 1 + x 2 α x α 1 + x α 2 for x 1, x 2 and α > 1, the second expressions for H in the theorem can be bounded from below by H [1 i 2 C i1 R i1 + C i2 R i2 α C i1 R i1 α C i2 R i2 α [. 29 α Cα i log C i To further bound the numerator in 29 we define the function fx = 1 + x α 1 x α cx α ɛ, where < ɛ < α 1, < c < 2 γ 1 and γ = α 1 ɛ. It can be shown that fx for x [, 1, since f = and f x αx γ 1 + 1/x γ 1 c on [, 1. Hence, by using the inequality fx, we derive for x 1, x 2, max{x 1, x 2 } > and x = min{x 1, x 2 }/max{x 1, x 2 } x 1 + x 2 α x α 1 x α 2 = max{x 1, x 2 } α 1 + x α 1 x α cmax{x 1, x 2 } α x α ɛ cmin{x 1, x 2 } α ; the inequality clearly holds even if max{x 1, x 2 } = since both of its sides are zero. Thus, by applying this last inequality to 29 and using 28, we obtain H c [1 i 2 min {C i1 R i1, C i2 R i2 } α [ α Cα i log C i This completes the proof. cδα P i 2, C i1 > δ, C i2 > δ[min{r i1, R i2 } α [ >. α Cα i log C i

21 Implicit Renewal Theory on Trees 21 Remarks: i ote that the assumptions [ of Theorem 4.2 differ slightly from those of Theorem 4.1 in the condition < Cα i log C i <, which is implied [ [ by Cα i log+ C i <, the strict convexity of ϕθ = Cθ i and the hypothesis that ϕ1 = ϕα = 1, as argued in the preceding proof. ii The assumption P Ñ 2 > is the minimal one to ensure the existence of a nontrivial solution, see conditions H in [22 and C4 in [2. Otherwise, when P Ñ 1 = 1, W n is a simple multiplicative random walk with no branching; clearly, in this case our expression for H reduces to zero. Also, if P C i = 1 = 1, R can only be a constant; see the remark above Theorem 2.1 in [23. However, this last case is excluded from the theorem since P C i = 1 = 1 implies C i 1 a.s., which, in conjunction with ϕα = 1, α > 1, yields P C i {, 1}, 1 i = 1, but this cannot happen due to the nonarithmetic assumption. iii ote also that condition C3 in [2 equivalent to P C i {, 1}, 1 i < 1 in our notation is implied by the nonarithmetic assumption of our theorem. Interestingly enough, if this last condition fails, Lemma 1.1 of [22 shows that equation 26 has no nontrivial solutions. iv As stated earlier, a version of this theorem was proved in Theorem 2.2 of [23, by transforming recursion 26 into a first order difference autoregressive/perpetuity equation on a different probability space, see Lemma 4.1 in [23. However, it appears that the method from [23 does not extend to the nonhomogeneous and non-linear problems that we cover here, since [ the proof of Lemma 4.1 in [23 critically depends on having both [R = 1 and C i = 1. Similarly as in Corollary 4.1, the constant H can be computed explicitly for integer α 2. Corollary 4.2. For integer α 2, and under the same assumptions of Theorem 4.2, the constant H can be explicitly computed as a function of [R k, [C k, 1 k α 1. In particular, for α = 2, H = [ j=i+1 C ic j [. C2 i log C i Proof. The proof follows directly from multinomial expansions of the second expression for H in Theorem 4.2. We also want to point out that for non-integer general α > 1 we can use Lemma 4.1 to obtain the following bound for H, [ R p 1 α/p 1 [ C i α H [, α Cα i log C i where p = α.

22 22 P.R. Jelenković and M. Olvera-Cravioto 5. The maximum recursion: R = C ir i Q In order to show the general applicability of the implicit renewal theorem, we study in this section the following non-linear recursion: R = D C i R i Q, 3 where Q,, C 1, C 2,... is a nonnegative random vector with { }, P Q > > and {R i } i is a sequence of iid random variables independent of Q,, C 1, C 2,... having the same distribution as R. ote that in the case of page ranking applications, where the {R i } represent the ranks of the neighboring pages, the potential ranking algorithm defined by the preceding recursion, determines the rank of a page as a weighted version of the most highly ranked neighboring page. In other words, the highest ranked reference has the dominant impact. Similarly to the homogeneous linear case, this recursion was previously studied in [6 under the assumption that Q, =, and the {C i } are real valued deterministic constants. The more closely related case of Q and {C i } being random was studied earlier in [18. Furthermore, these max-type stochastic recursions appear in a wide variety of applications, ranging from the average case analysis of algorithms to statistical physics; see [1 for a recent survey. Using standard arguments, we start by constructing a solution to 3 on a tree and then we show that this solution is finite a.s. and unique under iterations and some moment conditions, similar to what was done for the nonhomogeneous linear recursion in Section 4. Our main result of this section is stated in Theorem 5.1. Following the same notation as in Section 4, define the process V n = i A n Q i Π i, n, 31 on the weighted branching tree T Q,C, as constructed in Section 2. Recall that the convention is that Q,, C 1, C 2,... = Q,, C,1, C,2,... denotes the random vector corresponding to the root node. With a possible abuse of notation relative to Section 4, define the process {R n } n according to n R n = V k, n. k= Just as with the linear recursion from Section 4, it is not hard to see that R n satisfies the recursion R n = C,j R n 1 Q = C j R n 1 Q, 32 j where {R n 1 j } are independent copies of R n 1 corresponding to the tree starting with individual j in the first generation and ending on the nth generation. One can j

23 Implicit Renewal Theory on Trees 23 also verify that V n = D C,k Q k,...,in C k,...,ij = k=1 k,...,i n A n j=2 n C k V n 1,k, where {V n 1,k } is a sequence of iid random variables independent of, C 1, C 2,... and having the same distribution as V n 1. We now define the random variable R according to R lim n Rn = k=1 V k. 33 ote that R n is monotone increasing sample-pathwise, so R is well defined. Also, by monotonicity of R n, 32 and monotone convergence, we obtain that R solves R = C,j R j k= Q = C j R j Q, where {R j } j are iid copies of R, independent of Q,, C 1, C 2,.... Clearly this implies that R, as defined by 33, is a solution in distribution to 3. However, this solution might be. ow, we establish the finiteness of the moments of R, and in particular that R < a.s., in the following lemma; its proof uses standard contraction arguments but is included for completeness. [ Lemma 5.1. Assume that ρ β = < 1 and [Q β < for some β >. Cβ i Then, [R γ < for all < γ β, and in particular, R < a.s. Moreover, if β 1, R n L β R, where L β stands for convergence in β 1/β norm. Proof. By following the same steps leading to 11, we obtain that for any k, [ [ [V β k = = [Q β ρ k β. 34 Hence, [R β = i A k Q β i Πβ i [ k= V β k implying that [R γ < for all < γ β. L β [ k= i A k Q β i Πβ i V β k [Qβ 1 ρ β <, [ That R n R whenever β 1 follows from noting that [ R n R β k=n+1 V [ β k k=n+1 V β k and applying the preceding geometric bound for [V β k. Just as with the linear recursion from Section 4, we can define the process {R n} as R n R n 1 V n R, n 1,

24 24 P.R. Jelenković and M. Olvera-Cravioto where V n R = i A n R,iΠ i, 35 and {R,i } i U are iid copies of an initial value R, independent of the entire weighted tree T Q,C. It follows from 32 and 35 that R n+1 = C j R n 1 j n R,i C j,...,ik i A n,j k=2 Q = C j Rn,j Q, where {R n 1 j } are independent copies of R n 1 corresponding to the tree starting with individual j in the first generation and ending on the nth generation, and A n,j is the set of all nodes in the n + 1th generation that are descendants of individual j in the first generation. Moreover, {Rn,j } are iid copies of R n, and thus, Rn is equal in distribution to the process obtained by iterating 3 with an initial condition R. This process can be shown to converge in distribution to R for any initial condition R satisfying the following moment condition. Lemma 5.2. For any R, if [Q β, [R β < and ρ β < 1 for some β >, then R n R, with [R β <. Furthermore, under these assumptions, the distribution of R is the unique solution with finite β-moment to recursion 3. Proof. The result will follow from Slutsky s Theorem see Theorem 25.4, p. 332 in [1 once we show that V n R. To this end, recall that V n R is the same as V n if we substitute the Q i by the R,i. Then, for every ɛ > we have that P V n R > ɛ ɛ β [V n R β ɛ β ρ n β[r β by 34. Since by assumption the right-hand side converges to zero as n, then R n R. Furthermore, [R β < by Lemma 5.1. Clearly, under the assumptions, the distribution of R represents the unique solution to 3, since any other possible solution with finite β-moment would have to converge to the same limit. ow we are ready to state the main result of this section. Theorem 5.1. Let Q,, C 1, C 2,... be a nonnegative random vector, with { }, P Q > > and R be the solution to 3 given by 33. Suppose that there exists j 1 with P j, C j > > such that the measure P log C j du, [ C j >, j is nonarithmetic, and that for some α >, [Q α <, < [ Cα i log C i < and Cα i = 1. In addition, assume [ α 1. C i <,, if α > 1; or, [ 1+ɛ 2. Cα/1+ɛ i < for some < ɛ < 1, if < α 1.

25 Implicit Renewal Theory on Trees 25 Then, where H < is given by 1 H = [ Cα i log C i = [ P R > t Ht α, t, α C ir i Q α C ir i α [. α Cα i log C i [ v P α 1 R > v 1C i R > v dv Proof. By Lemma 5.1 we know that [R β < for any [ < β < α. The same arguments used in the proof of Theorem 4.1 give that Cγ i < for some γ < α. The statement of the theorem with the first expression for H will follow from Theorem 3.1 once we prove that condition 5 holds. Define R = C i R i Q. Then, [ P R > t 1C i R i > t P R > t P max C ir i > t 1 i [ + P max C ir i > t 1C i R i > t 1 i. Since R = D R max 1 i C i R i, the first absolute value disappears. The integral corresponding to the second term is finite by Lemma 4.6, just as in the proof of Theorem 4.1. To see that the integral corresponding to the first term, P R > t P max C ir i > t t α 1 dt, 1 i is finite we proceed as in the proof of Theorem 4.1. First we use Fubini s Theorem to obtain that P R > t P max C ir i > t t α 1 dt 1 i = 1 α [R α α max C ir i 1 i [ = 1 α α α C i R i Q α C i R i [Qα α.

26 26 P.R. Jelenković and M. Olvera-Cravioto ow, applying Theorem 3.1 gives P R > t Ht α, [ 1 [ where H = Cα j log C j v P α 1 R > v 1C jr > v dv. The same steps used in the proof of Theorem 4.1 give the second expression for H. 6. Other recursions and concluding remarks As an illustration of the generality of the methodology presented in this paper, we mention in this section other recursions that fall within its scope. One example that is closely related to the recursions from Sections 4 and 5 is the following R = D C i R i + Q, 36 where Q,, C 1, C 2,... is a nonnegative vector with { }, P Q > >, and {R i } i is a sequence of iid random variables independent of Q,, C 1, C 2,... having the same distribution as R. Recursion 36 was termed discounted tree sums in [1; for additional details on the existence and uniqueness of its solution see Section 4.4 in [1. Similarly as in [14, it appears that one could study other non-linear recursions on trees using implicit renewal theory. For example, one could analyze the solution to the equation R = C D i R i + B i Ri + Q, where Q,, C 1, C 2,... is a nonnegative vector with { }, P Q > >, and {R, R i } i 1 is a sequence of iid random variables independent of Q,, C 1, C 2,.... Here, the primary difficulty would be in establishing the existence and uniqueness of the solution as well as the finiteness of moments. 7. Proofs 7.1. Implicit renewal theorem on trees We give in this section the proof of Lemma 3.1. [ Proof of Lemma 3.1. Observe that the measure 1log C i du, C i > is nonarithmetic nonlattice by our assumption since, if we assume to the contrary that it is lattice on a lattice set L, then on the complement L c of this set we have [ = 1log C i L c, C i > P log C j L c, C j >, j >, which is a contradiction. Hence, η is nonarithmetic as well, and it places no mass at

27 Implicit Renewal Theory on Trees 27 due to the exponential term e αu. To see that it is a probability measure note that ηdu = = e αu 1log C j du = C α j Similarly, its mean is given by e αu 1log C j du = 1. uηdu = Cj α log C j. by Fubini s Theorem To show that µ n = η n we proceed by induction. Let F n denote the σ-algebra generated by { i, C i,1, C i,2,... : i A j, j n 1 }, F = σ, Ω, and for each i A n set V i = log Π i. Hence, using this notation we derive µ n+1, t = = t t = = = e αu i 1V i + log C i,j du i A n e αu i A n e αvi [ i A n t i A n e αvi η, t V i η, t vµ n dv, i 1V i + log C i,j du F n i e αu Vi 1log C i,j du V i F n where in the fourth equality we used the independence of i, C i,1, C i,2,... from F n. Therefore µ n+1 dt = η µ n dt and the induction hypothesis gives the result Moments of W n In this section we prove Lemmas 4.1, 4.2 and 4.3. We also include a result that provides bounds for [W p n for integer p, which will be used in the proof of Lemma 4.3. Proof of Lemma 4.1. Let p = β {2, 3,... } and γ = β/p β/β + 1, 1. Suppose first that k and define A p k = {j 1,..., j k k : j j k =

28 28 P.R. Jelenković and M. Olvera-Cravioto p, j i < p}. Then, for any sequence of nonnegative numbers {y i } i 1 we have k k y i β = y i pγ k = y p i + k y pγ i + j 1,...,j k A pk j 1,...,j k A pk γ p y j1 1 j 1,..., j yj k k k p y j1 1 j 1,..., j yj k k k γ, 37 k γ where for the last step we used the well known inequality x k i xγ i for < γ 1 and x i. We now use the conditional Jensen s inequality to obtain k β k C i Y i C i Y i β = j 1,...,j k A pk j 1,...,j k A pk j 1,...,j k A pk γ p C 1 Y 1 j1 C k Y k j k by 37 j 1,..., j k γ p C 1 Y 1 j1 C k Y k j k j 1,..., j k C 1,..., C k γ p [ C j1 1 j 1,..., j Cj k k Y j1 1 Y j k k C 1,..., C k. k Since {Y i } is a sequence of iid random variables having the same distribution as Y, independent of the C i s we have that [ [ Y j1 1 Y j k k C 1,..., C k = Y j1 1 Y j k k = Y j1 j 1 Y j k jk, where Y κ = [ Y κ 1/κ for κ 1 and Y 1. Since Y κ is increasing for κ 1 it follows that Y ji j i Y ji p 1. Hence Y j1 j 1 Y j k jk Y p p 1, which in turn implies that k β k γ C i Y i C i Y i β p C j1 1 j 1,..., j Cj k k Y p p 1 k j 1,...,j k A pk k p k γ = Y β p 1 C i C p i k β Y β p 1 C i.

29 Implicit Renewal Theory on Trees 29 This completes the proof for k finite. When k =, first note that from the well known inequality x 1 + x 2 β x β 1 + xβ 2 for x 1, x 2 and β > 1 we obtain the monotonicity in k of the following difference k+1 β k+1 C i Y i C i Y i β k β C i Y i k C i Y i β. Hence, β C i Y i C i Y i β = lim k k β C i Y i k C i Y i β 38 lim γ p C 1 Y 1 j1 C k Y k j k k j 1,..., j k j 1,...,j k A pk k β lim Y k β p 1 C i β = Y β p 1 C i, 39 where 38 and 39 are justified by monotone convergence. k β Proof of Lemma 4.2. We use the well known inequality y k i < β 1, y i, k, to obtain β [Wn β = C i W n 1,i [ C β i W β n 1,i = [W β n 1 ρ β by conditioning on, C i and Fubini s theorem ρ n β[w β iterating n times yβ i for = ρ n β[q β. 4 Before proving the moment inequality for general β > 1, we will show first the corresponding result for integer moments. [ Lemma 7.1. Let p {2, 3,... } and recall that ρ p = Cp i, ρ ρ 1. Suppose [ p [Q p <, C i <, and ρ ρ p < 1. Then, there exists a constant