Related papers: Fixed points of multivariate smoothing transforms …

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…

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,…

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…

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,…

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 $\xi=(\xi_{n})_{n\in\mathbb{N}}$ in time, which satisfies for…

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 $X\in \mathbb{R}^d$…

We consider smoothing equations of the form $$X ~\stackrel{\mathrm{law}}{=}~ \sum_{j \geq 1} T_j X_j + C$$ where $(C,T_1,T_2,\ldots)$ is a given sequence of random variables and $X_1,X_2,\ldots$ are independent copies of $X$ and independent…

Given any finite or countable collection of real numbers $T_j,j\in J$, we find all solutions $F$ to the stochastic fixed point equation \[W\stackrel{\mathrm {d}}{=}\inf_{j\in J}T_jW_j,\] where $W$ and the $W_j,j\in J$, are independent…

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…

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…

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…

Consider the multivariate smoothing transform fixed-point equation: $\eta =$ law of $ \sum_{i=1}^N A_i Z_i$, where $N \geq 0$ is a random integer, $(A_i)_{i \geq 1}$ are $d \times d$ random nonnegative matrices, $(Z_i)_{i \geq 1}$ is a…

This paper concerns piecewise-smooth maps on $\mathbb{R}^d$ that are continuous but not differentiable on switching manifolds (where the functional form of the map changes). The stability of fixed points on switching manifolds is…

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…

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…

In this work, using a new geometrical approach we study to the existence of the fixed-point of mappings that independence of the smoothness, and also of their single-values or multi-values. This work proved the theorems that generalize in…

Given a nonincreasing null sequence $T = (T_j)_{j \ge 1}$ of nonnegative random variables satisfying some classical integrability assumptions and $\mathbb{E}(\sum_{j}T_{j}^{\alpha})=1$ for some $\alpha>0$, we characterize the solutions of…

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…

We consider solutions of the stochastic equation $R=_d\sum_{i=1}^NA_iR_i+B$, where $N>1$ is a fixed constant, $A_i$ are independent, identically distributed random variables and $R_i$ are independent copies of $R$, which are independent…

We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant measure the stochastic fixed point problem. This generalizes…