Solutions to complex smoothing equations

Abstract

We consider smoothing equations of the form $$\begin{aligned} X ~\mathop {=}\limits ^{\text {law}}~ \sum _{j \ge 1} T_j X_j + C \end{aligned}$$ 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 of the sequence $$(C,T_1,T_2,\ldots )$$ . The focus is on complex smoothing equations, i.e., the case where the random variables $$X, C,T_1,T_2,\ldots $$ are complex-valued, but also more general multivariate smoothing equations are considered, in which the $$T_j$$ are similarity matrices. Under mild assumptions on $$(C,T_1,T_2,\ldots )$$ , we describe the laws of all random variables X solving the above smoothing equation. These are the distributions of randomly shifted and stopped Levy processes satisfying a certain invariance property called $$(U,\alpha )$$ -stability, which is related to operator (semi)stability. The results are applied to various examples from applied probability and statistical physics.