# Fixed points of the smoothing transform: two-sided solutions

@article{Alsmeyer2010FixedPO, title={Fixed points of the smoothing transform: two-sided solutions}, author={Gerold Alsmeyer and Matthias Meiners}, journal={Probability Theory and Related Fields}, year={2010}, volume={155}, pages={165-199} }

Given a sequence (C, T) = (C, T1, T2, . . .) of real-valued random variables with Tj ≥ 0 for all j ≥ 1 and almost surely finite N = sup{j ≥ 1 : Tj > 0}, the smoothing transform associated with (C, T), defined on the set $${\mathcal{P}(\mathbb R)}$$ of probability distributions on the real line, maps an element $${P \in \mathcal{P}(\mathbb R)}$$ to the law of $${C + \sum_{j \geq 1} T_j X_j}$$ , where X1, X2, . . . is a sequence of i.i.d. random variables independent of (C, T) and with…

## 59 Citations

The Smoothing Transform: A Review of Contraction Results

- Mathematics
- 2013

Given a sequence (C, T) = (C, T 1, T 2, …) of real-valued random variables, the associated so-called smoothing transform \(\mathcal{S}\) maps a distribution F from a subset Γ of distributions on…

Fixed points of multivariate smoothing transforms with scalar weights

- Mathematics
- 2015

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…

Solutions to complex smoothing equations

- Mathematics
- 2015

We consider smoothing equations of the form $$\begin{aligned} X ~\mathop {=}\limits ^{\text {law}}~ \sum _{j \ge 1} T_j X_j + C \end{aligned}$$X=law∑j≥1TjXj+Cwhere $$(C,T_1,T_2,\ldots )$$(C,T1,T2,…)…

Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations

- Mathematics
- 2015

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,…

Precise tail asymptotics of fixed points of the smoothing transform with general weights

- Mathematics
- 2015

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…

The fixed points of the multivariate smoothing transform

- Mathematics
- 2016

Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$(T1,T2,…) be a sequence of random $$d\times d$$d×d matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider…

Precise tail index of fixed points of the two-sided smoothing transform

- Mathematics
- 2013

We consider real-valued random variables R satisfying the distributional equation
$$\displaystyle{ R\stackrel{d}{=}\sum _{k=1}^{N}T_{ k}R_{k} + Q, }$$
where \(R_{1},R_{2},\ldots\) are iid…

Convergence of the population dynamics algorithm in the Wasserstein metric

- MathematicsElectronic Journal of Probability
- 2019

We study the convergence of the population dynamics algorithm, which produces sample pools of random variables having a distribution that closely approximates that of the {\em special endogenous…

On the derivative martingale in a branching random walk

- Mathematics
- 2020

We work under the A\"{\i}d\'{e}kon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate…

## References

SHOWING 1-10 OF 38 REFERENCES

Fixed Points of the Smoothing Transform: the Boundary Case

- Mathematics
- 2005

Let $A=(A_1,A_2,A_3,\ldots)$ be a random sequence of non-negative numbers that are ultimately zero with $E[\sum A_i]=1$ and $E \left[\sum A_{i} \log A_i \right] \leq 0$. The uniqueness of the…

Fixed points of the smoothing transformation

- Mathematics
- 1983

SummaryLet W1,..., WN be N nonnegative random variables and let
$$\mathfrak{M}$$
be the class of all probability measures on [0, ∞). Define a transformation T on
$$\mathfrak{M}$$
by letting Tμ be…

Elementary fixed points of the BRW smoothing transforms with infinite number of summands

- Mathematics
- 2003

A stochastic fixed point equation related to weighted branching with deterministic weights.

- Mathematics
- 2006

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…

Fixed points of a generalized smoothing transformation and applications to the branching random walk

- MathematicsAdvances in Applied Probability
- 1998

Let {A i : i ≥ 1} be a sequence of non-negative random variables and let M be the class of all probability measures on [0,∞]. Define a transformation T on M by letting Tμ be the distribution of ∑ i=1…

Symmetric fixed points of a smoothing transformation

- MathematicsAdvances in Applied Probability
- 2003

Let T = (T 1, T 2,…) be a sequence of real random variables with ∑ j=1 ∞ 1 |T j |>0 < ∞ almost surely. We consider the following equation for distributions μ: W ≅ ∑ j=1 ∞ T j W j , where W, W 1, W…

Fixed points of inhomogeneous smoothing transforms

- Mathematics
- 2012

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation , where means equality in distribution, is a given sequence of non-negative…

The functional equation of the smoothing transform

- Mathematics
- 2012

Given a sequence T = (Ti)i� 1 of non-negative random variables, a function f on the positive halfline can be transformed to E Q i� 1 f(tTi). We study the fixed points of this transform within the…