Heavy tailed solutions of multivariate smoothing transforms

Dariusz Buraczewski, Ewa Damek, Sebastian Mentemeier and Mariusz Mirek

Stochastic Processes and their Applications, 2013, vol. 123, issue 6, 1947-1986

Abstract: Let N>1 be a fixed integer and (C1,…,CN,Q) a random element of M(d×d,R)N×Rd. We consider solutions of multivariate smoothing transforms, i.e. random variables R satisfying R=d∑i=1NCiRi+Q where =d denotes equality in distribution, and R,R1,…,RN are independent identically distributed Rd-valued random variables, and independent of (C1,…,CN,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,…,CN).

Keywords: Distributions; General theory; Renewal theory; Branching (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2013.02.003

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