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Heavy tailed solutions of multivariate smoothing transforms

Author

Listed:
  • Buraczewski, Dariusz
  • Damek, Ewa
  • Mentemeier, Sebastian
  • Mirek, Mariusz

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).

Suggested Citation

  • Buraczewski, Dariusz & Damek, Ewa & Mentemeier, Sebastian & Mirek, Mariusz, 2013. "Heavy tailed solutions of multivariate smoothing transforms," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1947-1986.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:6:p:1947-1986
    DOI: 10.1016/j.spa.2013.02.003
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    References listed on IDEAS

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    1. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
    2. Caliebe, Amke & Rösler, Uwe, 2003. "Fixed points with finite variance of a smoothing transformation," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 105-129, September.
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    Cited by:

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    2. Jelenković, Predrag R. & Olvera-Cravioto, Mariana, 2015. "Maximums on trees," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 217-232.

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