A Stochastic Fixed Point Equation Related to Weighted Branching

For real numbers $C,T_{1},T_{2},...$ we find all solutions $\mu$ to the stochastic fixed point equation $W \sim\sum_{j\ge 1}T_{j}W_{j}+C$, where $W,W_{1},W_{2},...$ are independent real-valued random variables with distribution $\mu$ and $\sim$ means equality in distribution. All solutions are infinitely divisible. The set of solutions depends on the closed multiplicative subgroup of ${ R}_{*}={ R}\backslash\{0\}$ generated by the $T_{j}$. If this group is continuous, i.e. ${R}_{*}$ itself or the positive halfline ${R}_{+}$, then all nontrivial fixed points are stable laws. In the remaining (discrete) cases further periodic solutions arise. A key observation is that the Levy measure of any fixed point is harmonic with respect to $\Lambda=\sum_{j\ge 1}\delta_{T_{j}}$, i.e. $\Gamma=\Gamma\star\Lambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet-Deny theorem.


Introduction
Given a sequence C, T 1 , T 2 , ... of real-valued random variables with known joint distribution, consider the stochastic fixed point equation for i.i.d.real-valued random variables W, W 1 , W 2 , ... which are independent of (C, T ), T def = (T j ) j≥1 .The general goal is to determine all possible distributions of W such that (1.1) holds true.Every such solution is called a (distributional) fixed point.
Fixed points of (1.1) turn up in a natural way as limits of recursive equations of the form Here (T (n) , C (n) ), W 2 , ... are independent random variables for each n ≥ 0. The joint distribution of C (n) and 2 , ...) is given and converges to that of (C, T ).The W (n) j 's are copies of W (n) .The first study of a recursive system of type (1.2) appears in [26] for the sorting algorithm Quicksort which might still be the most prominent example.The method provided in [26] which applies to general divide and conquer algorithms is based on the contraction of the map K associated with (1.2) defined on the set of L p probability measures (p ≥ 1) endowed with the Mallows metric (see Section 2); [24], [27] and [30] may be consulted for good descriptions of the method, generalizations, more examples and accounts of relevant literature.[27] contains a systematic study of (1.1) as a contraction limit of (1.2).The results focus on conditions for the convergence of X (n) in the Mallows metric, the existence of solutions of (1.1), their moments and also their tail behavior.They all require at least second moments.
A very special case of equation (1.1), namely has been studied in insurance mathematics under the keyword "perpetuity", see [10], [12], and for related results in higher dimensions [19] (stochastic matrices) and [4] (random affine mappings in Hilbert space).(1.1) and (1.2) are intimately connected with weighted branching processes.For the simplest case, the Galton-Watson branching process (Z n ) n≥0 with offspring distribution (p j ) j≥0 and offspring mean m, (1.2) holds true upon setting W (n) = Z n , C (n) = 0 and T (n) = (1, ..., 1, 0, ...) with P( j≥1 T (n) j = k) = p k for all k, n ∈ N 0 .Replacing the 1's in T (n) with i.i.d.nonnegative random variables, we get a branching random walk which has been studied by many authors, see e.g.[2] and [3].In a general weighted branching process the weight of an individual is given as the weight of the mother times an independent random factor, see e.g.[28] for a further discussion including general C (n) .
Let us also mention a number of special examples, all having C = 0. Mandelbrot [23] introduced fractals and a class of models he called "canonique" which lead to equations of type (1.1).In this context, random measures and Hausdorff dimension attracted some attention, see [15] and [1] for results using contraction arguments.From a different perspective, Kahane and Peyriére [18] and Guivarc'h [16] considered positive solutions of (1.1) under the restriction of a fixed number of i.i.d.nonzero components T j of T .Motivated by questions related to infinite particle systems and again only for finitely many nonzero T j , Durrett and Liggett [11] provided a description of all nontrivial nonnegative fixed points of (1.1).Their approach relies on monotonicity properties of the Laplace transforms which cannot be used in case of general real-valued factors T j .Partial extensions of their results appear in [21] and [22] under the condition that the number of factors is random but still a.s.finite.
The purpose of this article is to study (1.1) for the case where (C, T ) is a vector of realvalued deterministic components C, T 1 , T 2 , ..., that is when C, T 1 , T 2 , ... are just constants.We will determine all fixed points.The major part will focus on the homogeneous case C = 0 and builds on an unpublished report by the second author [29].The case where C = 0 can then be treated rather shortly by drawing to a large extent on the findings in the homogeneous situation.In terms of the characteristic function ϕ of W equation (1.1) with C = 0 may be rewritten in the equivalent form It will be shown that, trivial cases excluded, any fixed point is infinitely divisible (Prop.4.1).
Hence we may take logarithms to obtain with Λ def = j≥1 1 {Tj =0} δ Tj and δ x denoting the Dirac measure at x.In a series of papers, Davies, Shimizu, Ramachandran, Rao, and others considered (1.4) for quite general Λ under various assumptions, see [25] for an account.They combine the integrated Cauchy functional equation with additional arguments.Our approach is similar, but uses a stronger form of the Choquet-Deny theorem involving characters and disintegration of measures (see the Appendix to this paper).This simplifies the structural arguments as well as the assumptions.
Here are some explicit examples.The normal distributions with mean 0 and variance σ 2 > 0 are the unique nonzero solutions to the equation see [9], [31] and [25].The distribution of a random variable W is called α-stable for α ∈ (0, 2] if, for all a, b > 0, there exists d ∈ R such that where W 1 , W 2 are independent copies of W and c > 0 is determined through , the solution is a stable distribution [9], [31], [25]. The condition on the generated group is necessary.Lévy (see [13, p.567]) gave an example of a symmetric non-stable distribution solving (1.6) in the special case i.e. a = b = 1 2 .In this case the smallest closed multiplicative group generated by a, b is obviously discrete.Corollary 7.5 will provide all symmetric solutions to (1.1) (and thus (1.6) with d = 0).In our notation used there Lévy's example corresponds to the case α = 1, We proceed with an outline of the main necessary steps that will furnish our main results to be presented in Section 7 (C = 0) and Section 9 (C = 0).Let C = 0 unless stated otherwise and let F denote the set of all distributional fixed points of (1.1).Section 2 provides some basic information on the map K associated with (1.1).K is defined on the set of probability measures on R (see (2.1)), and the elements of F are the fixed points of K. Trivial cases where all T j have modulus 0 or 1 are discussed in Section 3 and thus excluded from the subsequent analysis.Lemma 3.2 collects the then necessary conditions on T for the existence of nontrivial fixed points which are thus standing assumptions throughout the rest of the article.That all elements of F are infinitely divisible was already proved in [31] but will be reproved here for completeness by using the weighted branching representation of a fixed point (Section 4).The next step is to show that F contains nontrivial elements iff there exists an α ∈ (0, 2], in fact uniquely determined, such that j≥1 see Proposition 5.1.We call α the characteristic exponent of T .One may directly check that all symmetric α-stable distributions then belong to F as mentioned above already.However, there may be more fixed points.To determine F completely requires to account for the closed multiplicative subgroup of R * generated by Λ, that is G Λ .A key fact, first obtained for symmetric fixed points in the proof of Proposition 5.1 and then in Lemma 8.1 for any element of F, is that the Lévy measure Γ of such a fixed point satisfies where means multiplicative convolution.So Γ is Λ-harmonic which brings the powerful Choquet-Deny theorem [6] into play.This result tells us that any Λ-harmonic ν is a mixture over R * /G Λ of extremal measures of the form e(s −1 x)λ λ sGΛ (dx), where λ λ sGΛ denotes the Haar measure on the coset sG Λ def Though a rather abstract result in the general setting of locally compact Abelian groups (an Appendix collects some necessary general facts) it leads to a very explicit conclusion about Γ in the given situation because the set of characters satisfying (1.10) contains only one element, namely e(x) = |x| −α with α the characteristic exponent of T .We arrive at the conclusion that all nontrivial elements of F are stable laws or mixtures of certain periodic variants described in Section 6.Given this information the fixed point equation (1.1), more precisely its characteristic function version (1.4), boils down to an equation for the parameters of these distributions.
Our main results, which provide a complete description of F, are presented in Section 7.They must distinguish between five different cases because there are essentially five different closed multiplicative subgroups of R * that T can generate (listed in Section 5).If G Λ equals R * or R + , which is the most pleasant situation, all nontrivial elements of F are α-stable laws, thus normal distributions if α = 2.However, further fixed points of periodic type may occur if G Λ is discrete.Proofs are presented in Section 8.The case C = 0 is treated in Section 9.This can be done by a straightforward reduction to the homogeneous case (Theorem 9.2) whenever j≥1 T j ∈ R\{1}.If j≥1 T j equals 1 or does not exist in R there may be no fixed points at all (Theorem 9.3).The final Section 10 contains a brief discussion of the associated multiplicative random walk associated with T .The latter was used in [11] for the determination of all solutions to (1.1) for the case where C = 0 and T consists of a finite fixed number of positive random variables.

The fixed point equation and some properties
Let T j , j ≥ 1, be given real numbers and define the map K on the set of probability measures on R by where the random variables W 1 , W 2 , ... are independent with distribution µ and where L(X) means the distribution of a random variable X.The infinite sum need not exist but if it does it is understood in the sense of convergence in distribution of the finite sums n j=1 T j X j as n → ∞.Since these are sums of independent random variables the convergence actually holds true a.s., see e.g.[7, p. 292].Denote by D(K) the domain of K, i.e. the set of all probability distributions µ for which K(µ) is well defined.
As usual, denote by * the convolution of measures on the additive group R. The domain D(K) is closed under * because one can easily see that for all µ, ν ∈ D(K).K further commutes with the reflection operator R defined as R(µ) = L(−X) where X has distribution µ.So we have and the domain D(K) is closed under the operator R.
Our purpose is to completely describe the set of fixed points of K, that is of or, equivalently, the set F of associated characteristic functions.The point measure at 0 is a trivial fixed point.
Lemma 2.1.The set F is closed under convolution and reflection.
Proof.Given two fixed points µ, ν, we infer with the help of (2.2) and (2.3) that K(µ * ν) = K(µ) * K(ν) = µ * ν and KR(µ) = RK(µ) = µ, respectively.♦ As an immediate consequence we note: Corollary 2.2.If µ is a fixed point then the same holds true for its symmetrization In the following we will write the fixed point equation K(µ) = µ in terms of random variables, that is in the form (1.1) with C = 0: where W, W 1 , W 2 , ... have distribution µ.If denotes the characteristic function of W (or µ) then (2.4) is equivalent to The right hand side is necessarily well defined as the limit of n j=1 ϕ(T j t) by Lévy's continuity theorem.We note that the order of summation or multiplication in (2.4), respectively (2.5) is fixed and can in fact be crucial in cases where n j=1 T j converges to a finite limit while j |T j | = ∞.

Trivial cases
We already noted after the definition of F that µ = δ 0 is always a fixed point of K.However, there may be more such trivial solutions under special assumptions on T = (T 1 , T 2 , ...).We call µ ∈ F trivial if µ = δ c for some c in which case the characteristic function equals Proof.If all T j are 0 there is nothing to prove (F = {δ 0 }).Given the assumption of (a), the estimate implies that |ϕ(t)| is either 0 or 1 for every t, hence |ϕ| ≡ 1 because ϕ is continuous.This shows µ = δ c for some c.The proofs of (b) and (c) are easy and thus omitted.♦ Assuming that not all T j are in {−1, 0, 1}, we finally prove that the existence of a nontrivial fixed point not only implies that there must be nonzero T j with modulus = 1 but that in fact all nonzero T j must have modulus less than 1 and that there are at least two of them.Lemma 3.2.Suppose that not all T j are in {−1, 0, 1}.If there exists a nontrivial fixed Proof.Let ϕ be the characteristic function of a nontrivial fixed point µ.We first prove for some ε > 0 we infer |ψ| ≡ 1 on [−ε, ε] and then everywhere because ψ is a ch.f.(use Cor. 2 on p. 298 in [7]).This yields the contradiction |ϕ| ≡ 1.If all nonzero T j have modulus less than 1 choose an infinite subsequence (T j(k) ) k≥1 with |T j(k) | → 1.Then, by continuity, Finally, it is immediately seen that j≥1 1 {Tj =0} = 1 in combination with sup j≥1 |T j | < 1 implies F = {δ 0 } and thus that there can be no nontrivial solution.♦

The weighted branching representation and infinite divisibility
In this section we will show, under the assumption sup j≥1 |T j | < 1, that any fixed point of K is infinitely divisible.For this and later purposes we next give a brief description of the weighted branching process associated with equation (2.1).
Let V be the infinite tree with vertex set ∪ n≥0 N n where N 0 def = {∅}.Each vertex v = (v 1 , ..., v n ), which we also write as v 1 v 2 ...v n , is uniquely connected to the root ∅ by the path gives the total weight of the unique path from the root to v under multiplication.Now let X(v), v ∈ V, be i.i.d.random variables with common distribution µ and define for n ≥ 0. The notation |v|=n suggests that the order of summation does not matter.However, this may not be true because our assumptions will not guarantee that the L(v)X(v) are absolutely summable.In such cases we stipulate that the summation is to be understood in lexicographic order: |v|=n = v1≥1 ... vn≥1 .Indeed, this is the right summation when iterating equation (2.1), see (4.3) below.
(W n ) n≥0 forms a stochastic sequence called weighted branching process.It satisfies the backward equation where the W n,j = |v|=n L(jv) Tj X(jv), j ≥ 1, are i.i.d.copies of W n .This shows that K n (µ) = L(W n ) for each n ≥ 0. In particular, all W n have distribution µ if µ is a fixed point of K.
Under this assumption it is now easily seen that an n-fold iteration of equation (2.1) takes the form for every n ∈ N where the W (v), |v| = n, are i.i.d.copies of W .This representation will be used in the proof of Proposition 4.1 below.
A probability measure µ as well as its characteristic function ϕ (or a random variable X with L(X) = µ) is called infinitely divisible if for each n ∈ N there exists a characteristic function ϕ n such that ϕ = ϕ n n .Equivalently, µ can be decomposed as the n-fold convolution of a probability measure µ n having characteristic function ϕ n for each n.
A triangular scheme is an array of random variables Y j,k , j ∈ N and 1 ≤ k ≤ k j for some k j ∈ N. It is called independent if the rows consist of independent random variables, and it is called infinitesimal or asymptotically negligible if for any ε > 0. The connection with infinite divisibility is the following: The distributional limit of the row sums of any independent infinitesimal triangular scheme is necessarily inifinitely divisible (if the limit exists), see e.g.[14].This is the crucial fact to be used in the proof of the following result concerning the fixed points of K given by (2.1).Proof.Since trivial solutions are cleary infinitely divisible assume there is a nontrivial solution µ and suppose W d = µ.Then W satisfies (4.3) which implies that for each n ≥ 1 we find a finite set Hence we conclude the asserted infinite divisibility of µ from the result stated above.♦ The Lévy representation for an infinitely divisible distribution µ states that the logarithm of its characteristic function ϕ has a unique integral representation of the form identity function and Γ a not necessarily finite measure on R * , called Lévy measure associated with µ (or ϕ).Γ satisfies See [14] for different representations and [17] for an approach via Choquet representation theory.
In the situation of Proposition 4.1 where µ is a fixed point we further have for all u > 0 see [5,Lemma 6].Note that, if µ is symmetric, then γ = 0 and Γ is also symmetric whence 5. The characteristic exponent of T = (T 1 , T 2 , ...) Throughout the remainder of this article we always assume that T = (T 1 , T 2 , ...) satisfies This is justified because, by Lemma 3.2, nontrivial fixed points of K exist only under this condition when excluding the trivial cases discussed in Proposition 3.1.
It is positive because m(0) ≥ 2. We call this α the characteristic exponent of T hereafter and will next show that it always exists in (0, 2] whenever K has nontrivial fixed points.
Proposition 5.1.Suppose (A) and that F contains a nontrivial element.Then the characteristic exponent α of T exists and is an element of (0, 2]. Remark.If (A) holds but F = {δ c : c ∈ R} then the characteristic exponent α of T may be > 2 or not even exist.As an example for the first situation take and T i = 0, otherwise.Then j≥1 T j = 1 ensures δ c ∈ F for c ∈ R, while m(3) < 1 < m(2) implies 2 < α < 3.An example where α does not exist is given by T j = (−1) j+1 1 k for 2 k−1 ≤ j < 2 k and k ≥ 1.Indeed, j≥1 T j = 1 again holds true, but Before we can proceed with the proof some important facts must be collected.Recall that Λ = j≥1 1 {Tj =0} δ Tj and G Λ denotes the closed multiplicative subgroup of R * generated by Λ.There are five possible cases: (D2) G Λ = r Z for some r > 1.
Notice that condition (A) excludes the trivial subgroups {−1, 1} and {1}.The Haar measure (unique up to multiplicative constants), denoted as λ λ GΛ hereafter, equals |u| −1 du in the continuous cases (C1) and (C2), and counting measure in the discrete ones.Let E(G Λ ) be the set of characters of G Λ , that is the set of all continuous positive functions e : G Λ → R + satisfying e(xy) = e(x)e(y) for all x, y ∈ G Λ .Of particular interest for our purposes is the subset It is not difficult to check that in all five cases the characters are given by the functions Moreover, we infer upon noting e l (x −1 ) Λ(dx) = j≥1 |T j | l = m(l) that, under (A), E 1 (Λ) is either void or consists of the single element e α , α the characteristic exponent of T .Now consider a Radon measure µ on R * and suppose that µ is Λ-harmonic, defined by µ = µ Λ.Here means multiplicative convolution, that is for any measurable f : The set of all Λ-harmonic measures is a convex cone.By the Choquet-Deny theorem which we state in an Appendix at the end of the paper we infer that any nonzero Λ-harmonic µ has a unique integral representation where µ e (dx) def = e(x)λ λ GΛ (dx) for e ∈ E and µ is a finite measure on E 1 (Λ) × R * /G Λ endowed with the Baire σ-field.If E 1 (Λ) = ∅ there is no Λ-harmonic measure.Otherwise, E 1 (Λ) = {e α } so that µ must equal c(δ eα ⊗ μ) for some probability measure μ on the factor group R * /G Λ and a c > 0. This means that for some c 1 , c 2 ≥ 0 with c 1 + c 2 > 0.
Proof of Proposition 5.1.Let µ be a nontrivial fixed point with characteristic function ϕ.Since µ ∈ F implies µ s = µ * R(µ) ∈ F (Corollary 2.2) it is no loss of generality to assume that µ is symmetric with nonnegative ϕ.Using (2.5) and the Lévy representation (4.7) we infer the equation for all t ∈ R with unique σ 2 ≥ 0 and Lévy measure Γ.Note that the last expression exists as the limit of the nondecreasing sequence 1 2 σ 2 t 2 n j=1 T 2 j + n j=1 (1 − cos(T j tu)) Γ(du), n ∈ N. We conclude that either σ 2 > 0 in which case m(2) = j≥1 T 2 j = 1 (thus α = 2), or σ 2 = 0 and (5.5) reduces to for all t ∈ R, in particular for sufficiently small ε > 0. With the help of (4.6), we also have By combining this with (5.7) we see that Γ Λ is also a Lévy measure, and the uniqueness of Γ entails Γ = Γ Λ (5.8) and therefore, by (5.2), that Γ(du) = c y u α λ λ yGΛ (du) Γ(dy) (5.9) for some c > 0 and a probability measure Γ on R * /G Λ .The number α > 0 is the unique characteristic exponent of T , and it satisfies α < 2 because in any of the five cases for G Λ mentioned above.♦

Stable and sG-stable distributions
An infinitely divisible distribution µ is called stable if the set is closed with respect to additive convolution * .This is just an equivalent formulation of property (1.6) stated in the Introduction.The characteristic function ϕ of a stable distribution has the representation where α ∈ (0, 2] is the so-called index of µ, β ∈ [−1, 1], γ ∈ R, c ≥ 0 are further parameters, sgn(t) denotes the sign of t (sgn(0) def = 0), and The Lévy measure Γ of µ is the null measure in case α = 2 and equals otherwise.Here c 1 , c 2 are nonnegative number satisfying c 1 + c 2 > 0 and The representation (6.2) is unique unless α = 2 or c = 0 in which case β is arbitrary.
For the continuous cases (C1) and (C2) where the subgroup G Λ generated by Λ is uncountable we will see in the next section that all nontrivial fixed points of K are stable distributions.However, fixed points of a more general type occur when G Λ is discrete (cases (D1-3)).The following notion of a sG-stable distribution provides an appropriate class for these additional solutions.Here G is an infinite closed multiplicative subgroup of R * , s an element of the factor group R * /G and sG the usual coset {sx : x ∈ G}.Recall that λ λ sG equals |u| −1 1 sG (u)du if G is continuous and counting measure on sG if G is discrete.Definition 6.1.Given G and s as just stated, an infinitely divisible distribution µ with characteristic function ϕ is called sG-stable with index α ∈ (0, 2) if for some γ ∈ R and c > 0.
One can immediately check that if G equals R * itself and thus s = 1 then the G-stable distributions are just the ordinary symmetric stable distributions with index α ∈ (0, 2).If G = R + then s ∈ {−1, 1} and the sG-stable distributions are the one-sided stable distributions concentrated either on R + or R − .However, if G is one of the discrete subgroups listed in (D1-3) the sG-stable distributions are no longer stable.On the other hand, the set defined in (6.1), but with a ∈ r N , is again closed under additive convolution for any sG-stable distribution µ.Furthermore, if X d = µ then rX is rsG-stable with rs computed modulo G.Note finally that an sG-stable distribution is symmetric iff γ = 0 and G = −G, thus G = R * or G = r Z ∪ −r Z for some r > 1.In this case (6.4) simplifies to log ϕ(t) = c (cos(su) − 1)|u| −α λ λ G (du) (6.5) for some c > 0.
In the discrete cases (D1-3) mixtures of sG Λ -stable distributions will arise as additional solutions of the fixed point equation K(µ) = µ.

Main Results
We are now in the position to present our main results.The following theorems provide a full description of all nontrivial solutions to K(µ) = µ in the possible cases (C1-2) and (D1-3).
They are given in terms of their charactersistic function which amounts to a description of F. Condition (A) will be in force throughout and α always denotes the characteristic exponent of T .Let us stipulate hereafter that j≥1 T j = 1 can mean lim n→∞ n j=1 T j = 1 or that this limit does not exist at all.
i.e. the nontrivial fixed points of K are exactly the stable laws of index α with γ = 0.
i.e. the nontrivial fixed points of K are exactly the normal distributions with mean 0.
The next theorem provides a complete description of the fixed points of K (in terms of their characteristic functions) for discrete G Λ , but without a distinction of the three cases (D1-3).A specialization to these follows in a subsequent corollary.Let us note that we have Here means isomorphic equality.
where the probability measure ν on R * /G Λ and the constants c ≥ 0, γ ∈ R are subject to the following contraint: If in case j≥1 T j = 1, and in case j≥1 T j = 1.So the nontrivial fixed points of K are the normal distributions if j≥1 T j = 1, and the centered normal distributions otherwise.
Let F be the set of all triples (γ, c, ν) for which (7.3) holds true.Then, for the discrete case, Theorem 7.3 provides us in principle with a complete description of F (or F) in terms of F (which is one to one unless c = 0 in which case the triple (γ, 0, ν) pertains to ϕ(t) = e iγt regardless of ν).On the other hand, the appearing condition (7.3) naturally demands for further examination.Doing so while considering the cases (D1-3) separately, one is led to a more explicit description of F stated as Theorem 7.4 below.
Let M Λ be the set of probability measures ν on R * /G Λ (when identified with the subsets of R * given before Theorem 7.3) and M s Λ the subset of symmetric ν.Note that M s Λ is empty in Case (D1) because then R * /G Λ ⊂ R + .For the Cases (D2) and (D3) we further need the class of symmetric sG Λ -stable distributions (including trivial solutions where c = 0).(b) Suppose (D2) holds in which case all T j are nonnegative.If (ii) α = 1, then for each pair (c, ν) ∈ [0, ∞) × M Λ there exists exactly one γ such that (γ, c, ν) ∈ F.
(c) Suppose (D3) holds.If (i) α = 1, or α = 1 and j T j = j sgn(T j )|T j | α , then (ii) α = 1 and j T j = 1, then F = R × [0, ∞) × M 0 Λ .(iii) α = 1 and j T j exists in R ∪ {±∞} but does not equal j sgn(T j )|T j | α or 1, then for each pair (c, ν) ∈ (0, ∞) × M Λ there exists exactly one γ such that (γ, c, ν) ∈ F. (iv) α > 1 and j T j does not exist, then Remark.It should be clear that the fixed points provided in Theorem 7.1 for the case G Λ = R * (under respective conditions on α and T ) remain to be fixed points for any discrete subgroup G Λ .They are obtained when choosing ν as the uniform distribution on R * /G Λ .If G Λ is also a subgroup of R + (Case (D2)) the same holds true for the fixed points given in Theorem 7.2.These are obtained by choosing ν as a mixture of uniform distributions on the two congruent connected components of R * /G Λ .
A description of the set F s of symmetric fixed points of K or, equivalently, the associated set Fs of characteristic functions is easily derived from the previous results and thus summarized without proof in the subsequent corollary.As for the discrete cases, we only note that ϕ of the form (7.1) belongs to Fs iff γ = 0 and at least one of λ λ GΛ and ν is symmetric.Plainly, λ λ GΛ is symmetric in the case (D2) where G Λ = r Z ∪ −r Z for some r > 1.
i.e. the nontrivial symmetric fixed points are the mixtures of symmetric sG Λ -stable distributions.(d) If α ∈ (0, 2) and G Λ equals r Z or (−r) Z for some r > 1, then i.e. the nontrivial symmetric fixed points are the symmetric mixtures of sG Λ -stable distributions.
Remark.Let us point out that in cases where j T j exists in R while j |T j | = ∞, the set F may depend on the order of summation of the T j .It is indeed a well known fact in such a situation that for each x ∈ R we can find a rearrangement T π(1) , T π(2) , ... satisfying j T π(j) = x.But our results show that the set of fixed points belonging to a rearrangement π with j T π(j) = 1 generally differs from the corresponding set when j T π(j) = x = 1.

Proofs
In the proof of Proposition 5.1 we were led to the conclusion that the Lévy measure Γ of any nontrivial symmetric fixed point µ satisfies Γ = Γ Λ where denotes multiplicative convolution, see (5.8).Lemma 8.1(c) shows this be true for any nontrivial fixed point and provides us with the key to determine Γ by an application of the powerful Choquet-Deny theorem (see (5.9) for symmetric µ).
Proof.The "only if-part" is easily obtained by checking that any ϕ that meets the conditions in (a-c) satisfies the fixed point equation (1.4).So we can immediately proceed with the proof of the "if-part".
Obviously, trivial fixed points must satisfy the asserted conditions so that we may focus on nontrivial ones.Given the characteristic function ϕ of any nontrivial fixed point with Lévy-Khinchine representation (4.4), let Γ s be the Lévy measure of its symmetrization µ s .We have Γ s (du) = Γ(du) + Γ(−du) and Γ s = Γ s Λ, the latter being true by (5.8) because µ s is symmetric.It follows that for all n ≥ 1.By combining (2.5) with the Lévy-Khinchine representation of ϕ we get Consequently, γ n also converges to some γ ∈ R as n → ∞.Returning to (8.3) for all t ∈ R. The uniqueness of the Lévy-Khinchine representation finally implies that γ = γ , σ 2 = 0 or m(2) = 1, and Γ = Γ Λ. ♦ Proof of Theorem 7.1.It is easily checked that all elements of F as asserted pertain to fixed points of K for the respective cases.So we must conversely show that there are no other ones.To that end let µ be any fixed point of K with characteristic function ϕ having Lévy-Khinchine representation (4.4).Since G Λ = R * and Γ is Λ-harmonic, we have by (5.3) that Γ(du) = c|u| −α−1 du for some c ≥ 0.
If α < 2 then σ 2 = 0 by Lemma 8.1(b) and Γ(du) = c|u| −α−1 du for some positive c because µ is nontrivial.We infer that µ is a stable distribution with index α and characteristic function ϕ(t) = exp(itγ − c|t| α ), see (6.2).Again, using (2.5) and the Lévy-Khinchine representation we have and therefore tγ = tγ j T j for all t ∈ R. The uniqueness of the representation implies that γ( j T j − 1) = 0 and thus γ = 0 unless j≥1 T j = 1.Note that j≥1 T j < j≥1 |T j | ≤ 1 for α ≤ 1.Now one can easily check that µ is of the asserted type.♦ Proof of Theorem 7.2.The proof in case G Λ = R + is very similar to the previous one and we therefore restrict ourselves to a few comments.Again we must only verify that a nontrivial solution µ is of the type asserted in the theorem for the respective cases.By (5.4), its Lévy measure Γ this time has the form Γ(du we conclude Γ ≡ 0 by the same argument as above, while c 1 + c 2 > 0 must hold if α ∈ (0, 2).The nonnegativity of the T j together with the uniqueness of α as a solution to j≥1 T α j = 1 implies j≥1 T j = 1 whenever α = 1.For the case α = 2 this entails that only the centered normal distributions can be fixed points (γ = 0).In all other cases a fixed point must be a stable law, and the uniqueness of the Lévy-Khinchine representation may once again be employed to arrive at the asserted constraints of the parameters.Further details are omitted.♦ Proof of Theorem 7.3.We only consider the case α ∈ (0, 2) because for α = 2 the Lévy measure of any nontrivial fixed point equals again 0. After this observation the remaining arguments are the same as in the previous two theorems.
If α ∈ (0, 2) then, by (5.9), the Lévy measure Γ of any nontrivial fixed point equals for some probability measure ν on R * /G Λ and some c > 0. We have σ 2 = 0 by Lemma 8.1(b).Hence for each s ∈ R * /r Z Case (D3).Suppose G Λ = (−r) Z for some r > 1.It is easily checked with the help of (8.4) (though not directly seen upon inspection) that F s,c ≡ 0 in case α 2 ).Since F s,c is odd in s the following computation is only done for the case s > 0. We infer with (8.4)

The inhomogeneous case
In this section we will briefly discuss the fixed point equation (1.1) with a nonzero constant C (inhomogeneous case).So the map K is now defined as with independent X 1 , X 2 , ... having common distribution µ.The weighted branching representation of any fixed point for all n ≥ 1 and is obtained by successive iteration of (1.1).The W (v), v ∈ V, are independent copies of W . Summation may again be a subtle point.Recall from Section 4 that |v|=n = v1≥1 ... vn≥1 .If we iterate (1.1) once we get and thus see that the summands going with C by rearrangement may be separated from those going with the W (ij). On the other hand, we cannot conclude at this point that the limit in (9.3) exists when taken for both terms n i=1 j≥1 L(ij)W (ij) and C n i=1 T j separately.In particular, it is not clear at this point whether or not j T j exists in R when there is a fixed point (see Theorem 9.3 for an answer).These remarks apply, of course, to all higher order iterations of (1.1) as well.
Refraining from a discussion of trivial cases we assume from the beginning that T satisfies condition (A) hereafter.By (9.2) and a similar argument as in the proof of Proposition 4.1, we see that each nontrivial fixed point µ of K can be obtained as the limit of an independent infinitesimal triangular scheme and is thus infinitely divisible.So the logarithm of its characteristic function ϕ exists everywhere and satisfies (compare (1.4)) log ϕ(t) = iCt + j≥1 log ϕ(T j t). (9.4) Embarking on this observation the following lemma is just the straightforward extension of as n → ∞.
Let F C be the set of fixed points of K for general C, hence F = F 0 .If j T j exists in R and does not equal 1, then we can provide a simple description of F C in terms of F 0 for which we may resort to the results in Section 7. Put Theorem 9.2.If T satisfies (A) and if j T j ∈ R\{1}, then Proof.It suffices to note that W solves equation (1.1) with C = 0 iff (under the given assumptions) which means that W + C(T ) is a fixed point for the homogeneous equation (C = 0).♦ Left with the cases that j T j equals 1 or does not exist in R, the following two results provide complete answers.Theorem 9.3 shows in particular that the existence of j T j in R constitutes a necessary condition for F C = ∅ (for any C = 0).Theorem 9.3.The set F C is empty for each C = 0 whenever j T j does not exist in R, or when j T j = 1 in one of the cases (C1), (D1), or α = 2 holds true.
Proof.Suppose C = 0 and that j T j equals 1 or does not exist in R. The Lévy measure Γ of any fixed point must satisfy Γ = Γ Λ (Lemma 9.1).In each of the three cases (C1), (D1), and α = 2 we have seen in Section 8 that this in turn implies the symmetry of Γ, in particular (xχ(u) − χ(xu)) Γ(du) = 0. Hence we infer F C = ∅ because condition (9.5), which simplifies to C + γ n j=1 T j → γ, is clearly impossible to satisfy.It is a matter of checking (9.4) or (9.5) to arrive at the same conclusion whenever j T j does not exist in R (and no further condition on α or G Λ ).We omit the details.♦ It remains to look at the cases (C2), (D2) and (D3) for any T additionally satisfying j T j = 1.The assertions of (b) and (c) are immediately obtained when observing that equation (9.5) is equivalent to γ = C + j γT j + F s,c (T j ) ν(ds) (compare (7.3)) which in turn leads to (8.11), respectively (8.12) with C added on the right hand side.Further details can be omitted.♦ 10 The associated random walk for any real-valued f for which the expectation exists.
Proof.The assertions are easily verified when using the independence structure in the weighted branching model described in Section 4. The result appears also in [3,Lemma 4.1] for the case where T consists of a random number of i.i.d.nonzero random variables (branching random walk case).We therefore omit the details.♦ Returning to the fixed point equation for all t ∈ R * .

Theorem 7 . 3 .
(G Λ discrete).Let the smallest closed multiplicative subgroup generated by the T j be one of the discrete subgroups listed in (D1-3).
(a) If α ∈ (0, 2) then F consists of all ϕ of the form

F 1 ( 8
which proves the assertion including B(s) > 0, if α < 1, and B(s) < 0 if α > 1.If α = 1 itsuffices to note that the middle term of the third line in the above computation simplifies to−r ζ−m m−l−1 n=−l r (1−α)n = −r ζ−m m = sx log r x,while the first and last one cancel each other.♦ Given one of the cases (D2) or (D3), the previous lemma does now easily lead to an expression for n j=1 F s,c (T j ) that will enable us to prove Theorem 7.4.We obtain: Case (D2).If G Λ = r Z for some r > 1, then all T j are nonnegative and n j=1 s,c (T j ) = c n j=1 k∈Z r −αk A(T j , sr k ) sgn(s)|s| α n j=1 (T j − T α j ), if α = 1, cs n j=1 T j log r T j , if α =

Lemma 8 .Lemma 9 . 1 .
1 to general C ∈ R. Let ϕ be the characteristic function of any infinitely divisible distribution µ with Lévy-Khinchine representation (4.4).Then µ is a fixed point of K iff the conditions (a-c) in Lemma 8.1 hold true, where (8.1) is modified to