A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks

Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.], arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons in [Lyons, R.(1997). A simple path to Biggins' martingale convergence for branching random walk. In Athreya, K.B., Jagers, P. (eds.). Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, vol. 84, Springer, Berlin, 217-221.]. We also provide a necessary and sufficient condition for uniform integrability of these martingales in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.


Introduction and results
The principal purpose of this article is to provide a log-type moment result for the limit of iterated i.i.d. random linear functions, called perpetuties. It is given as Theorem 1.2 in the following subsection along with all necessary facts about the model. A similar result (Theorem 1.4) will then be formulated for the a.s. limit of a well-known martingale associated with the branching random walk introduced in Subsection 1.2. As will be explained in Section 5, the connection between these at first glance unrelated models pops up when studying the weighted random tree associated with the branching random walk under the so-called size-biased measure. It does not take by surprise that this connection, once established, can be utilized to obtain moment results in the branching model by resorting to corresponding ones for perpetuities.

The random discounted sum
obtained as the a.s. limit of Z n under appropriate conditions (see Proposition 1.1 below), is called perpetuity and of interest in various fields of applied probability like insurance and finance, the study of shot-noise processes or, as will be seen further on, of branching random walks. The law of Z ∞ appears also quite naturally as the stationary distribution of the (forward) iterated function system where Ψ n (t) def = Q n +M n t for n = 1, 2, ... and Φ 0 is independent of {(M n , Q n ) : n = 1, 2, ...}. Due to the recursive structure of this Markov chain, it forms solution of the stochastic fixed point equation where as usual the variable Φ is assumed to be independent of (M, Q). Let us finally note that Z ∞ may indeed be obtained as the a.s. limit of the associated backward system when started at Φ 0 ≡ 0, i.e.
Goldie and Maller [13] gave the following complete characterization of the a.s. convergence of the series in (1.1). For x > 0, define

Condition (1.4) holds particularly true if
E log |M| ∈ (−∞, 0) and E log + |Q| < ∞, (1.7) and for this special case results on the finiteness of certain log-type moments of Z ∞ were derived in [15] and [17]. To extend those results to the general situation with (1.3) being the only basic assumption is one purpose of the present paper. Let the function b : R + → R + be measurable, locally bounded and regularly varying at ∞ with exponent α > 0. Functions b of interest in the following result are, for instance, b(x) = x α log k x or b(x) = x α exp(β log γ x) for β ≥ 0, 0 < γ < 1 and k ∈ N, where log k denotes k-fold iteration of the logarithm.
Replacing lim n→∞ Π n = 0 a.s. with the stronger condition E log |M| ∈ (−∞, 0), this result is stated as Theorem 3 in [15], and our proof also fixes a minor flaw appearing in the proof given there.
Since (1.8) and (1.9) are conditions in terms of the absolute values of M and Q, the first conclusion of Theorem 1.2 remains valid when replacing (1.10) with the stronger assertion If Π n → 0 a.s. and if Z ∞ and Z * ∞ are both a.s. finite and nondegenerate, this leads us to the conclusion that (1.10) and (1.11) are actually equivalent. A similar conclusion has been obtained in [2] for the case of ordinary moments (viz. b(log x) = x p for some p > 0), see Theorem 1.4 there.

The branching random walk and its intrinsic martingales
In the following we give a short description of the standard branching random walk, its intrinsic martingales and an associated multiplicative random walk.
Consider a population starting from one ancestor located at the origin and evolving like a Galton-Watson process but with the generalization that individuals may have infinitely many children. All individuals are residing in points on the real line, and the displacements of children relative to their mother are described by a point process gives the total number of offspring of the considered mother and X i the displacement of the i-th child. The displacement processes of all population members are supposed to be independent copies of Z. We further assume Z({−∞}) = 0 and EN > 1 (supercriticality) including the possibility P{N = ∞} > 0 as already stated above. If P{N < ∞} = 1, then the population size process forms an ordinary Galton-Watson process. Supercriticality ensures survival of the population with positive probability.
For n = 0, 1, ... let Z n be the point process that defines the positions on R of the individuals of the n-th generation, their total number given by Z n (R). The sequence {Z n : n = 0, 1, ...} is called branching random walk (BRW). Let and edges connecting each v ∈ V with its successors vi, i = 1, 2, ... The length of v is denoted as |v|. Call v an individual and |v| its generation number. A BRW {Z n : n = 0, 1, ...} may now be represented as a random labeled subtree of V with the same root. This subtree T is obtained recursively as follows: For any v ∈ T, let N(v) be the number of its successors (children) and denote the point process describing the displacements of the children vi of v relative to their mother. By assumption, the Z(v) are independent copies of Z. The Galton-Watson tree associated with this model is now given by and X i (v) denotes the label attached to the edge (v, vi) ∈ T×T and describe the displacement of vi relative to v. Let us stipulate hereafter that |v|=n means summation over all vertices of T (not V) of length n.
Suppose there exists γ > 0 such that For n = 1, 2, ..., define F n def = σ(Z(v) : |v| ≤ n − 1), and let F 0 be the trivial σ-field. Put Notice that the dependence of W n on γ has been suppressed. The sequence {(W n , F n ) : n = 0, 1, ...} forms a non-negative martingale with mean one and is thus a.s. convergent with limiting variable W , say, satisfying EW ≤ 1. It has been extensively studied in the literature, but first results were obtained in [21] and [5]. Note that P{W > 0} > 0 if, and only if, {W n : n = 0, 1, ...} is uniformly integrable. While uniform integrability is clearly sufficient, the necessity hinges on the well known fact that W satisfies the stochastic fixed point equation (1.14) for n = 1, 2, ..., where the W (v), |v| = n, are i.i.d. copies of W that are also independent of {L(v) : |v| = n}, see e.g. [7]. In fact W (v) is nothing but the a.s. limit of the martingale { |w|=m Our goal is to study certain moments of W in the nontrivial situation where {W n : n = 0, 1, ...} is uniformly integrable. For the latter to hold, Theorem 1.3 below provides us with a necessary and sufficient condition, again under no additional assumptions on the BRW beyond (1.12). In order to formulate it, we first need to introduce a multiplicative random walk associated with our model. Let M be a random variable with distribution defined by for any Borel subset B of R + . Notice that the right-hand side of (1.15) does indeed define a probability distribution because E |v|=1 L(v) = EW 1 = 1.
More generally, we have (see e.g. [7], Lemma 4.1) for each n = 1, 2, ..., whenever {M k : k = 1, 2, ...} is a family of independent copies of M and Π n def = n k=1 M k . It is important to note that P{M = 0} = 0 and P{M = 1} < 1. (1.17) The first assertion follows since, by (1.15), P{M > 0} = EW 1 = 1. As for the second, observe that P{M = 1} = 1 implies E |v|=1 L(v)1 {L(v) =1} = 0 which in combination with EW 1 = 1 entails that the point process Z consists of only one point u with L(u) = 1. This contradicts the assumed supercriticality of the BRW. Not surprisingly, the chosen notation for the multiplicative random walk associated with the given BRW as opposed to the notation in the previous subsection is intentional, and we also keep the definitions of J(x) and A(x) from there, see (1.2) and thereafter. and There are three distinct cases in which conditions (1.18) and (1.19) hold simultaneously: (A1) E log M ∈ (−∞, 0) and EW 1 log For the case (A1), Theorem 1.3 is due to Biggins [5] and Lyons [25], see also [22]. In the present form, the result has also been stated as Proposition 1 in [17] (with a minor misprint), however without proof and a reference to the proof of Theorem 2 in [14] instead. But the latter theorem was formulated in terms of fixed points rather than martingale convergence which somewhat obscures how to extract the necessary arguments. On the other hand, the study of uniform integrability has a long history, going back to the famous Kesten-Stigum theorem [20] for ordinary Galton-Watson processes and the pioneering work by Biggins [5] for the BRW, and followed later by work in [23] and [25]. We have therefore decided to include a complete (and rather short) proof here. The existence of moments of W was studied in quite a number of articles, see [3], [5], [9], [15], [17], [24], [26]. The following theorem, which is our second main moment-type result, goes beyond the afore-mentioned ones in that it does not restrict to case (A1) of Theorem 1.3. The function b(x) occurring here is of the type stated before Theorem 1.2.
An interesting aspect of this theorem is that it provides conditions for the existence of Φ-moments of W for Φ slightly beyond L 1 without assuming the (LlogL)-condition to ensure uniform integrability. The latter condition is a standing assumption in a related article by Alsmeyer and Kuhlbusch [3] where a similar but more general result (as for the functions Φ) is proved, see Theorem 1.2 there.
There are basically two probabilistic approaches towards finding conditions for the existence of EΦ(W ) for suitable functions Φ. The method of this paper, worked out in [14] and [17], hinges on getting first a momenttype result for perpetuities (here Theorem 1.2) and then translating it into the framework of branching random walks. This is accomplished by an appropriate change of measure argument (see the proof of Theorem 1.3). The second approach, first used in [4] for Galton-Watson processes and further elaborated in [3], relies on the observation that BRW's bear a certain double martingale structure which allows the repeated application of the convex function inequalities due to Burkholder, Davis and Gundy (see e.g. [11]) for martingales. Both approaches have their merits and limitations. Roughly speaking, the double martingale argument requires as indispensable ingredients only that Φ be convex and at most of polynomial growth. On the other hand, it also comes with a number of tedious technicalities caused by the repeated application of the convex function inequalities. The basic tool of the method used here is only Jensen's inequality for conditional expectations, but it relies heavily on the existence of a nonnegative concave function Ψ that is equivalent at ∞ to the function Φ(x)/x. This clearly imposes a strong restriction on the growth of Φ.
The rest of the paper is organized as follows. Section 2 collects the relevant properties of the functions involved in our analysis, notably b(x), b(log x) and A(x), followed in Section 3 by some preliminary work needed for the proofs of Theorems 1.2 and 1.4. In particular, a number of moment results for certain functionals of multiplicative random walks are given there which may be of independent interest (see Lemma 3.5). Theorem 1.2 is proved in Section 4, while Section 5 contains the proofs of Theorems 1.3 and 1.4.

Properties of the functions involved
for all n ≥ 1, where f ∼ g has the usual meaning that lim x→∞ f (x)/g(x) = 1. By Lemma 1 in [1], b(x) may further be chosen in such a way that for all x, y ∈ R + and some C ∈ (0, ∞). The smoothness of b(x) (and thus of ℓ(x)) and property (2.1) will be standing assumptions throughout without further notice. Before giving a number of lemmata, let us note the obvious facts that for any fixed a > 0.
Lemma 2.1. There exist smooth nondecreasing and concave functions f and .
for all x, y ∈ R + and a suitable constant C ∈ (0, ∞).
Proof. We must check lim x→∞ φ(xy)/φ(x) = 1 for y > 1. By the previous lemma, which yields the desired conclusion because f (x) ∼ b(log x) is slowly varying and, by (P3), J(log x+log y) ∼ J(log x) for any fixed y. The second assertion follows as a simple consequence so that we turn directly to (2.4). Fix K ∈ N so large that (2.5) Note next that J as a nondecreasing sublinear function satisfies J(x + y) ≤ C(J(x) + J(y)) for all x, y ∈ R + . By combining this with the monotonicity of f, J and inequality (2.2), we obtain if x > K and y > K (thus xy > K)

Auxiliary results
In the notation of Subsection 1.1 and always assuming (1.3), let us consider the situation where |Z ∞ | < ∞ and the nondegeneracy condition (1.6) is in force. Then lim n→∞ Π n = 0 by Proposition 1.1, and holds true for each m ≥ 1, where (setting Π k:l ∞ constitutes a copy of Z ∞ independent of (M 1 , Q 1 ), ..., (M m , Q m ). We thus see that Z ∞ may also be viewed as the perpetuity generated by i.i.d. copies of (Π m , Q (m) ) for any fixed m ≥ 1. We may further replace m by any a.s. finite stopping time σ to obtain ∞ is a copy of Z ∞ independent of σ and {(M n , Q n ) : 1 ≤ n ≤ σ} (and thus of (Π σ , Q (σ) )). For our purposes, a relevant choice of σ will be σ def = inf{n ≥ 1 : |Π n | ≤ 1}, (3.4) which is nothing but the first (weakly) ascending ladder epoch for the random walk S n def = − log |Π n |, n = 0, 1, ... Ef sup
for a suitable C ∈ (0, ∞), where subadditivity of f has been used for the first inequality and (2.2) for the second one. Finally, we must verify (3.6). With m 0 denoting a median of Z ∞ , Goldie and Maller (see [13], p. 1210) showed that and this gives the desired conclusion by (3.8) and the fact that f is subadditive and satisfying (2.2). Remark 3.2. Let Q n be a conditional symmetrization of Q n given M n . Then a tail inequality similar to (3.9) holds for sup k≥1 |Π k−1 Q k | as well. However, in contrast to the Q (2) k , the Q k may be degenerate in which case an analog of (3.10) does not follow. This is the reason for considering sup k≥1 |Π 2k−2 Q (2) k | in the above lemma.
Proof. We first note that the moment assumption and lim x→∞ f (x) = ∞ together ensure sup n≥1 Π n−1 Q n < ∞ a.s. Therefore, there exists an η > 1 such that α = P{sup n≥1 Π n−1 Q n ≤ η} > 0. We further point out that the monotonicity of f and (2.2) imply f (Q 1/2 ) ≥ Cf (Q/2) for some C ∈ (0, 1). Now fix any γ > η and infer for x ≥ η (with V as in the previous lemma) the last inequality following by Lemma 3.3. With this at hand, we further and this proves the assertion because f (x)J(log x) is slowly varying at infinity by Lemma 2.3. Ef |M| J log + |M| < ∞. (3.14) where b is regularly varying with index α > 0. A proof for the special case b(x) = x α can be found in [19], as mentioned above. But the arguments given there are easily seen to hold for regularly varying b as well whence further details are omitted here. " for n = 0, 1, ... The random vectors ( M n , Q n ), n = 1, 2, ... are independent copies of ( M , Q) As, by construction, P{ M ≤ 1} = 1 and P{ M = 1} = 0, Lemma 3.4 enables us to conclude that Ef (sup n≥0 |Π n |) = Ef (sup n≥1 Π n−1 Q n ) < ∞ implies Ef Q J log + Q < ∞ which is the desired result. Finally, "(3.17)⇒(3.14)" follows from the obvious inequality sup 0≤n<σ |Π n | ≥ |M 1 | ∨ 1 and the fact that f (x)J(log x) is nondecreasing.
We will consider these cases separately, in fact Case (C2) will be handled by reducing it to the first case via an appropriate stopping argument.
Using the properties of f stated in Lemma 2.1 (which particularly ensure subadditivity) and sup n≥0 |Π n | = |Π 0 | = 1, we obtain for fixed a ∈ (0, 1) The second integral is easily estimated with the help of (3.11) as so that we are left with an estimation of I 1 .

Case (C2):
As already announced, we will handle this case by using a stopping argument based on the ladder epoch σ given in (3.4). We adopt the notation of the proof of Lemma 3.5, in particular (σ n ) n≥0 denotes the sequence of successive ladder epochs associated with σ. Put further Q n def = σn k=σ n−1 +1 |Π σ n−1 +1:k−1 Q k | for n ≥ 1 which are independent copies of Q def = Q 1 = Q (σ) . Notice that It will be shown now that condition (4.1) holds true with Q instead of Q.
Since M = |Π σ | ∈ (0, 1) a.s. and thus satisfies the condition of Case (C1), we then arrive at the desired conclusion Ef (Z * ∞ ) < ∞. By Lemma 2.2, there is a subadditive φ(x) of the same asymptotic behavior as f (x)J(log x), as x → ∞. Hence it suffices to verify Eφ( Q) < ∞. Use the obvious inequality in combination with property (2.4) and the subadditivity of φ to infer .
But the right hand expression is finite because Eφ( Q) < ∞ is ensured by (1.8) and Lemma 3.5 and because follows from Wald's identity, condition (1.9) and Eσ < ∞ which in turn is a consequence of our assumption lim n→∞ Π n = 0 a.s.
Left with the proof of (1.8), we get Ef (sup n≥0 |Π n |) < ∞ by another appeal to Lemma 3.1 and then the assertion by invoking Lemma 3.5. This completes the proof of Theorem 1.2.
5 Size-biasing and the results for W n

Modified branching random walk
We adopt the situation described in Subsection 1.2. Recall that Z denotes a generic copy of the point process describing the displacements of children relative to its mother in the considered population. The following construction of the associated modified BRW with a distinguished ray (v n ) n≥0 , called spine, is based on [8] and [25].
Let Z * be a point process whose law has Radon-Nikodym derivative m(γ) −1 i=1 e γX i with respect to the law of Z. The individual v 0 = ∅ residing at the origin of the real line has children, the displacements of which relative to v 0 are given by a copy Z * 0 of Z * . All the children of v 0 form the first generation of the population, and among these the spinal successor v 1 is picked with a probability proportional to e γs if s is the position of v 1 relative to v 0 (size-biased selection). Now, while v 1 has children the displacements of which relative to v 1 are given by another independent copy Z * 1 of Z * , all other individuals of the first generation produce and spread offspring according to independent copies of Z (i.e., in the same way as in the given BRW). All children of the individuals of the first generation form the second generation of the population, and among the children of v 1 the next spinal individual v 2 is picked with probability e γs if s is the position of v 2 relative to v 1 . It produces and spreads offspring according to an independent copy Z * 2 of Z * whereas all siblings of v 2 do so according to independent copies of Z, and so on. Let and P(( W n ) n≥0 ∈ ·) = Q((w n ) n≥0 ∈ ·). (5. 2) The relevance of these definitions with view to the martingale {W n : n = 0, 1, ...} to be studied hereafter is provided by the following lemma (see Prop. 12.1 and Thm. 12.1 in [8]).
Lemma 5.1. For each n ≥ 0, w n is the Radon-Nikodym derivative of Q with respect to P on G n . Moreover, if w def = lim sup n→∞ w n , then (1) w n is a P-martingale and 1/w n is a Q-martingale.
(2) EW = E P w = 1 if and only if P{ W < ∞} = Q{w < ∞} = 1. Let us point out that, in view of (5.1) and (5.2), the first assertion of Lemma 5.1(1) just restates the martingale property of W n , while the second one says that the same holds true for 1/ W n . The link between W n and W n is provided by Lemma 5.2. For each n = 0, 1, ..., W n is a size-biasing of W n , that is for each function f : R + → R + . More generally, for each Borel function h : (R + ) n+1 → R + .
Proof. It suffices to note that, by Lemma 5.1, for each n = 0, 1, ... and h as stated, where the G n -measurability of (w 0 , ..., w n ) should be observed for the second equality.

Connection with perpetuities
Next we have to make the connection with perpetuities. For u ∈ T, let N (u) denote the set of children of u and, if |u| = k, Since all individuals off the spine reproduce and spread as in the unmodified BRW, we have that the { W n (u) : n = 0, 1, ...} for u ∈ n≥0 N (v n )\{v n+1 } are independent copies of {W n : n = 0, 1, ...}. For n ∈ N, define further Then it is easily checked that the {(M n , Q n ) : n = 1, 2, ...} are i.i.d. with distribution given by for any Borel set A, where (M, Q) denotes a generic copy of (M n , Q n ) and our convention |u|=n = u∈Tn should be recalled from Section 1. In particular, for any measurable B, that is Notice that this implies As for the distribution of M, we have which is in accordance with the definition given in (1.15). As we see from (5.5), Here is the lemma that provides the connection between the sequence { W n : n = 0, 1, ...} and the perpetuity generated by {(M n , Q n ) : n = 0, 1, ...}. Let A be the σ-field generated by {(M n , Q n ) : n = 0, 1, ...} and the family {Z * n : n = 0, 1, ...}, where Z * n is the copy of Z * describing the displacement of the children of v n relative to its mother. For n ≥ 1 and k = 1, ..., n, put also and notice that E R n,k |A = 0 because each W n−k (u) is independent of A with mean one.
Lemma 5.3. With the previous notation the following identities hold true for each n ≥ 0: and Proof. Each v ∈ T n has a most recent ancestor in {v k : k = 0, 1, ...}. By using this and recalling (5.6) and (5.9), one can easily see that which obviously gives (5.10) as Π 0 = 1. But the second assertion is now immediate when observing that E(Π k−1 R n,k |A) = Π k−1 E(R n,k |A) = 0 a.s.

Two further auxiliary results
We continue with two further auxiliary results about the martingale W n and its size-biasing W n .
Lemma 5.4. Let W * def = sup n≥0 W n and W * def = sup n≥0 W n . Then, for each a ∈ (0, 1), there exists b = b(a) ∈ R + such that for all t > 1 P{W > t} ≤ P{W * > t} ≤ b P{W > at}. (5.12) As a consequence for any non-negative nondecreasing concave function f . Replacing (W, W * ) with ( W , W * ), the same conslusions hold true (with b/a instead of b).
Proof. (5.12) is due to Biggins [6] for the case of a.s. finite branching (see Lemma 2 there) and has been obtained without this restriction as Lemma 1 in [16] by a different argument. Its counterpart for ( W , W * ) can be found as Lemma 3 in [17], but the following argument (for the nontrivial part) using (5.12) and Lemma 5.2 is more natural and much shorter: for all t > 1.
Proof. The first statement follows, as W 1 = 1 a.s. implies the same for each W n , n ≥ 2 (use W n = |v|=n−1 L(v)W 1 (v) with independent W 1 (v) which are copies of W 1 and independent of the L(u), |u| = n − 1). Conversely, if W (and thus also W as its size-biasing) is degenerate, then the fixed point equation (1.14) for n = 1 combined with EW = 1 yields Π k−1 Q k < ∞ a.s. and thus lim inf n→∞ W n < ∞ a.s. As {1/ W n : n = 0, 1, ...} constitutes a positive and thus a.s. convergent martingale (see after Lemma 5.1), we further infer W = lim inf n→∞ W n and thereupon the desired P{ W < ∞} = 1.
Remark 5.6. With view to the subsequent proof of Theorem 1.4 it is useful to point out that the previous proof has shown that, if {W n : n = 0, 1, ...} is uniformly integrable, W = lim n→∞ W n < ∞ a.s. and Consequently, if f : R + → R + denotes any nondecreasing and concave function, then an application of Jensen's inequality (for conditional expectations) in combination with ( Sufficiency. Let Z ∞ be defined as usual with M k and Q k as in (5.5) and (5.6), respectively. Notice that Z * ∞ = k≥1 |Π k−1 Q k | = Z ∞ in the present context. By Lemma 5.2 and (5.7), condition (1.20) translates to Eb(log + W 1 )J(log + W 1 ) = Eb(log + Q)J(log + Q) < ∞, and we may naturally replace b(log + x) with the asymptotically equivalent concave function f from Lemma 2.1. Since we also infer Ef (M)J(log + M) < ∞. Hence the desired conclusion (1.21), equivalently EW f (W ) < ∞, follows by an appeal to Theorem 1.2 and (5.15).