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Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment

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  • Li, Yingqiu
  • Liu, Quansheng
  • Peng, Xuelian

Abstract

For Mandelbrot’s cascade in a random environment, we find the critical value for the existence of harmonic moments of the limit variable of Mandelbrot’s martingale, and establish large and moderate deviation principles for the free energy. As applications, we show the corresponding limit theorems for branching random walks in random environments.

Suggested Citation

  • Li, Yingqiu & Liu, Quansheng & Peng, Xuelian, 2019. "Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 57-65.
    • Handle: RePEc:eee:stapro:v:147:y:2019:i:c:p:57-65
    DOI: 10.1016/j.spl.2018.10.002
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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.
    2. Gao, Zhiqiang & Liu, Quansheng, 2016. "Exact convergence rates in central limit theorems for a branching random walk with a random environment in time," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2634-2664.
    3. Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    4. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.
    5. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
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