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Heavy tailed solutions of multivariate smoothing transforms

Sebastian Mentemeier, Dariusz Buraczewski, Ewa Damek, Mariusz Mirek
published 2012-06-11

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 $(C_1,..., C_N)$.

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