Given a sequence $T=(T_i)_{i\geq1}$ of nonnegative random variables, a function f on the positive halfline can be transformed to $\mathbb{E}\prod_{i\geq1}f(tT_i)$. We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Laplace transforms of probability distributions on $[0,\infty)$, the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, $X\stackrel{d}{=}\sum_{i\geq1}T_iX_i$, where $\stackrel{d}{=}$ denotes equality of the corresponding laws, and $X_1,X_2,...$ is a sequence of i.i.d. copies of X independent of T. Further, since left-continuous survival functions are covered as well, the results also apply to the fixed-point equation $X\stackrel{d}{=}\inf\{X_i/T_i:i\geq1,T_i>0\}$. Moreover, we investigate the phenomenon of endogeny in the context of the smoothing transform and, thereby, solve an open problem posed by Aldous and Bandyopadhyay.
Let $N,d > 1$ be fixed integers, let $(T_1, ..., T_N)$ be random d-by-d matrices with nonnegative entries and $Q$ a random d-vector with nonnegative entries. This induces a mapping (the multivariate smoothing transform) on probability laws on the nonnegative cone by $S η:= \mathrm{Law\ of}\ (T_1 X_1 + ... + T_N X_N + Q)$, where the $X_i$ are iid with law $η$ and independent of $(T_1, ..., T_N, Q)$. Under conditions similar to those for the well-studied case d=1, a complete characterization of all fixed points of $S$ is obtained.
Given a nonincreasing null sequence $T = (T_j)_{j \ge 1}$ of nonnegative random variables satisfying some classical integrability assumptions and $\mathbb{E}(\sum_{j}T_{j}^α)=1$ for some $α>0$, we characterize the solutions of the well-known functional equation $$ f(t)\,=\,\textstyle\mathbb{E}\left(\prod_{j\ge 1}f(tT_{j})\right),\quad t\ge 0, $$ related to the so-called smoothing transform and its min-type variant. In order to do so within the class of nonnegative and nonincreasing functions, we provide a new three-step method whose merits are that (1) it simplifies earlier approaches in some …
We consider complex Mandelbrot multiplicative cascades on a random weigh\-ted tree. Under suitable assumptions, this yields a dynamics $\T$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations) and which have a finite moment of order 2. Moreover, we can exhibit two main behaviors: If the weights are conservative, i.e., sum up to~1 almost surely, we find a domain for the initial law $μ$ such that a non-standard (functional) central limit theorem is valid for the orbit $(\T^nμ)_{n\ge 0}$ (this completes in a non trivial way our previous result in th…
We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation $X \stackrel{d}{=} C + \sum_{i \geq 1} T_i X_i$, where $\stackrel{d}{=}$ means equality in distribution, $(C,T_1,T_2,...)$ is a given sequence of non-negative random variables and $X_1,X_2,...$ is a sequence of i.i.d.\ copies of the non-negative random variable $X$ independent of $(C,T_1,T_2,...)$. In this situation, $X$ (or, more precisely, the distribution of $X$) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necess…
We deal with the equation $Y \stackrel{\rm d}{=} \frac{1}{b} \sum_{1\le j\le N} W_jY_j$, where the unknown is the distribution of $Y$, the variables in the right hand side are independent, the $Y_j$ are equidistributed with $Y$, $N$ is an integer valued random variable, and the $W_j$ are equidistributed, nonnegative and of expectation~1. Usually a solution is obtained as the limit of a martingale. In some cases we give an explicit formula for the law of $Y$.
Let $N > 1$ be a fixed integer and $(C_1,..., C_N,Q)$ a random element of $GL(d, \R)^N x \R^d$. We consider solutions of multivariate smoothing transforms, i.e. random variables $R$ satisfying $$R \eqdist \sum_{i=1}^N C_i R_i +Q $$ where $\eqdist$ denotes equality in distribution, and $R, R_1,..., R_N$ are independent identically distributed $\R^d$-valued random variables, and independent of $(C_1,..., C_N, Q)$. We briefly review conditions for the existence of solutions, and then study their asymptotic behaviour. We show that under natural conditions, these solutions exhibit heavy tails. Our …
Given a sequence $(C_1,\ldots,C_d,T_1,T_2,\ldots)$ of real-valued random variables with $N := \#\{j \geq 1: T_j \not = 0\} < \infty$ almost surely, there is an associated smoothing transformation which maps a distribution $P$ on $\mathbb{R}^d$ to the distribution of $\sum_{j \geq 1} T_j \mathbf{X}^{(j)} + \mathbf{C}$ where $\mathbf{C} = (C_1,\ldots,C_d)$ and $(\mathbf{X}^{(j)})_{j \geq 1}$ is a sequence of independent random vectors with distribution $P$ independent of $(C_1,\ldots,C_d,T_1,T_2,\ldots)$. We are interested in the fixed points of this mapping. By improving on the techniques devel…
Given a sequence $(T_1, T_2, ...)$ of random $d \times d$ matrices with nonnegative entries, suppose there is a random vector $X$ with nonnegative entries, such that $ \sum_{i \ge 1} T_i X_i $ has the same law as $X$, where $(X_1, X_2, ...)$ are i.i.d. copies of $X$, independent of $(T_1, T_2, ...)$. Then (the law of) $X$ is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case $d=1$, a function $m$ is introduced, such that the existence of $α\in (0,1]$ with $m(α)=1$ and $m'(α) \le 0$ guarantees the existence of nontrivial fixed points. …
Given a sequence $(C,T) = (C,T_1,T_2,...)$ of real-valued random variables with $T_j \geq 0$ for all $j \geq 1$ and almost surely finite $N = \sup\{j \geq 1: T_j > 0\}$, the smoothing transform associated with $(C,T)$, defined on the set $\mathcal{P}(\R)$ of probability distributions on the real line, maps an element $P\in\mathcal{P}(\R)$ to the law of $C + \sum_{j \geq 1} T_j X_j$, where $X_1,X_2,...$ is a sequence of i.i.d.\ random variables independent of $(C,T)$ and with distribution $P$. We study the fixed points of the smoothing transform, that is, the solutions to the stochastic fixed-p…
At each time $n\in\mathbb{N}$, let $\bar{Y}^{(n)}=(y_{1}^{(n)},y_{2}^{(n)},\cdots)$ be a random sequence of non-negative numbers that are ultimately zero in a random environment $ξ=(ξ_{n})_{n\in\mathbb{N}}$ in time, which satisfies for each $n\in\mathbb{N}$ and a.e. $ξ,~E_ξ[\sum_{i\in\mathbb{N}_{+}}y_{i}^{(n)}(ξ)]=1.$ The existence and uniqueness of the non-negative fixed points of the associated smoothing transform in random environments is considered. These fixed points are solutions of the distributional equation for $a.e.~ξ,~Z(ξ)\overset{d}{=}\sum_{i\in\mathbb{N}_{+}}y_{i}^{(0)}(ξ)Z_{i}(Tξ…
This paper is devoted to the study of the stochastic fixed-point equation X \stackrel{d}{=} \inf_{i \geq 1: T_i > 0} X_i/T_i and the connection with its additive counterpart $X \stackrel{d}{=} \sum_{i\ge 1}T_{i}X_{i}$ associated with the smoothing transformation. Here $\stackrel{d}{=}$ means equality in distribution, $T := (T_i)_{i \geq 1}$ is a given sequence of nonnegative random variables and $X, X_1, ...$ is a sequence of nonnegative i.i.d. random variables independent of $T$. We draw attention to the question of the existence of nontrivial solutions and, in particular, of special solution…
The branching random walk (BRW) smoothing transform $T$ is defined as $T:\text{distr}(U_{1})\mapsto \text{distr} (\sum_{i=1}^{L}X_{i}U_{i})$, where given realizations $\{X_{i}\}_{i=1}^{L}$ of a point process, $U_{1},U_{2},...$ are conditionally independent identically distributed random variables, and $0\leq \text{Prob}\{L=\infty \}\leq 1$. Given $α\in (0,1]$, $α$-\emph{elementary} fixed points are fixed points of $T$ whose Laplace-Stieltjes transforms $ϕ$ satisfy $\underset{s\to +0}{\lim}\dfrac{1-ϕ(s)}{s^α}=m$, where $m$ is any given positive number. If $α=1$, these are the fixed points with …
For a given random sequence $(C,T_{1},T_{2},\ldots)$ with nonzero $C$ and a.s. finite number of nonzero $T_{k}$, the nonhomogeneous smoothing transform $\mathcal{S}$ maps the law of a real random variable $X$ to the law of $\sum_{k\ge 1}T_{k}X_{k}+C$, where $X_{1},X_{2},\ldots$ are independent copies of $X$ and also independent of $(C,T_{1},T_{2},\ldots)$. This law is a fixed point of $\mathcal{S}$ if the stochastic fixed-point equation (SFPE) $X\stackrel{d}{=}\sum_{k\ge 1}T_{k}X_{k}+C$ holds true, where $\stackrel{d}{=}$ denotes equality in law. Under suitable conditions including $\mathbb{E}…
Given $d \ge 1$, let $(A_i)_{i\ge 1}$ be a sequence of random $d\times d$ real matrices and $Q$ be a random vector in $\mathbb{R}^d$. We consider fixed points of multivariate smoothing transforms, i.e. random variables $X\in \mathbb{R}^d$ satisfying $X$ has the same law as $\sum_{i \ge 1} A_i X_i + Q$, where $(X_i)_{i \ge 1}$ are i.i.d. copies of $X$ and independent of $(Q, (A_i)_{i \ge 1})$. The existence of fixed points that can attract point masses can be shown by means of contraction arguments. Let $X$ be such a fixed point. Assuming that the action of the matrices is expanding as well wit…
We consider real-valued random variables R satisfying the distributional equation R \eqdist \sum_{k=1}^{N}T_k R_k + Q, where R_1,R_2,... are iid copies of R and independent of T=(Q, (T_k)_{k \ge 1}). N is the number of nonzero weights T_k and assumed to be a.s. finite. Its properties are governed by the function m(s) := \E \sum_{k=1}^N |T_k|^s . There are at most two values α< βsuch that m(α)=m(β)=1. We consider solutions R with finite moment of order s > α. We review results about existence and uniqueness. Assuming the existence of βand an additional mild moment condition on the T_{k}, our ma…
