Solutions to complex smoothing equations

Abstract

We consider smoothing equations of the form $$X\stackrel{\mathrm{law}}{=}\sum _{j\ge 1}{T}_j{X}_j+C$$ where $(C,T_1,T_2,\dots )$ is a given sequence of random variables and $X_1,X_2,\dots$ are independent copies of $X$ and independent of the sequence $(C,T_1,T_2,\dots )$. The focus is on complex smoothing equations, i.e., the case where the random variables $X,C,T_1,T_2,\dots$ are complex-valued, but also more general multivariate smoothing equations are considered, in which the $T_j$ are similarity matrices. Under mild assumptions on $(C,T_1,T_2,\dots )$, we describe the laws of all random variables $X$ solving the above smoothing equation. These are the distributions of randomly shifted and stopped Lévy 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.

67 pages

References in corpus (2)

• Fixed points of multivariate smoothing transforms with scalar weights
• The space requirement of m-ary search trees: distributional asymptotics for m >= 27

Cited by in corpus (1)

• Convergence of complex martingales in the branching random walk: the boundary