Heavy tailed solutions of multivariate smoothing transforms

2012/06/30 by Dariusz Buraczewski, Ewa Damek, Sebastian Mentemeier +1 · 1 citation
Mathematics · #math.PR #msc:60E05 #msc:60K05 #msc:60J80

paper · pdf · doi:10.1016/j.spa.2013.02.003

published as Stochastic Processes and their Applications 123(6) pp. 1947-1986 (2013) · 35 pages

arxiv created 2013/02/05 · arxiv updated 2013/04/04

Abstract

Let N > 1 be a fixed integer and (C₁,..., 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 ∑_i=1^N C_i R_i +Q where \eqdist denotes equality in distribution, and R, R₁,..., R_N are independent identically distributed \R^d-valued random variables, and independent of (C₁,..., 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 (C₁,..., C_N).

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