Heavy tailed solutions of multivariate smoothing transforms

TL;DR

This paper investigates the heavy-tailed behavior of solutions to multivariate smoothing transforms, establishing conditions under which solutions exhibit heavy tails, including cases with complex-valued weights.

Contribution

It provides new insights into the asymptotic heavy-tailed behavior of solutions to multivariate smoothing equations, extending to complex weights.

Findings

Solutions exhibit heavy tails under natural conditions.

Results include complex-valued weights.

Conditions for existence and asymptotic behavior are characterized.

Abstract

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 $\eqdist$ denotes equality in distribution, and $R, R_1,..., R_N$ are independent identically distributed $\R^d$-valued random variables, and independent of $(C_1,..., C_N, Q)$. We briefly review conditions for the existence of solutions, and then study their asymptotic behaviour. We show that under natural conditions, these solutions exhibit heavy tails. Our results also cover the case of complex valued weights $(C_1,..., C_N)$.