Fixed points of the smoothing transform: classical results and recent developments 

Speaker: Matthias Meiners（University of Giessen, Germany）

Title: Fixed points of the smoothing transform: classical results and recent developments 

Inviter: 随机分析研究中心

Language: English 

Time & Venue: 2026 年9月4日15:00-16:00 南楼613

Abstract: Subject of my talk are smoothing equations of the form \begin{equation*}    \textstyle

\mathcal{L}(X) = \mathcal{L}\left(\sum_{j=1}^N T_j X_j + C\right) \end{equation*}

where $\mathcal{L}(Y)$ denotes the law of a random variable $Y$, and $N$ is a random nonnegative integer. Solutions to the above smoothing equation, also known as the fixed-point equation of the smoothing transformation, arise as limiting distributions of various quantities of interest in models from applied probability and statistical physics. I will survey classical results on this equation in dimension $d=1$, first for nonnegative random variables and then for real-valued random variables.

I will subsequently turn to results in various multivariate settings, including ongoing work on the case in which $X,X_1,X_2,\ldots$ are independent and identically distributed $d$-dimensional random vectors, $C$ is a given $d$-dimensional random vector, and $T_1,T_2,\ldots,T_N$ are given random elements of the general linear group of degree $d$.