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Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform

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  • Mei, Jianzhang
  • Liu, Quansheng

Abstract

Consider the multivariate smoothing transform fixed-point equation: Z=∑i=1NAiZi in law, where N ≥ 0 is a random integer, (Ai)i ≥ 1 are d × d random nonnegative matrices, (Zi)i ≥ 1 is a sequence of independent R+d-valued random variables, independent of (N,A1,A2,…), each Zi having the same law as Z. For each fixed point, under suitable conditions, we describe its support, establish its absolute continuity, and prove the existence of its harmonic moments. The absolute continuity of Rd-valued fixed point in the not necessarily homogeneous case is also considered.

Suggested Citation

  • Mei, Jianzhang & Liu, Quansheng, 2026. "Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform," Stochastic Processes and their Applications, Elsevier, vol. 197(C).
    • Handle: RePEc:eee:spapps:v:197:y:2026:i:c:s0304414926000670
    DOI: 10.1016/j.spa.2026.104935
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    References listed on IDEAS

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    1. Liu, Quansheng, 1999. "Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 61-87, July.
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    3. Liu, Quansheng, 2001. "Asymptotic properties and absolute continuity of laws stable by random weighted mean," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 83-107, September.
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    6. 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.
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