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Cramér Moderate Deviations and Berry-Esseen Bounds for Mandelbrot’s Cascade in a Random Environment

Author

Listed:
  • Yingqiu Li

    (Changsha University of Science and Technology)

  • Peihan Li

    (Changsha University of Science and Technology)

  • Yushao Wei

    (Changsha University of Science and Technology)

Abstract

Let $$(Y_n)$$ ( Y n ) be a Mandelbrot’s cascade in an independent and identically distributed (i.i.d.) random environment $$\xi $$ ξ . According to the existence of the annealed Laplace transform of the limit variable $$W = {\lim _{n \rightarrow \infty }}{W_n}$$ W = lim n → ∞ W n , where $${W_n} = {{{Y_n}} / {{E_\xi }}}{Y_n}$$ W n = Y n / E ξ Y n is the normalized population size, and with the use of the associated random walks, Cramér moderate deviations and Berry-Esseen bounds for $$\log Y_n$$ log Y n are established. It is shown that harmonic moments of Mandelbrot’s martingale $$(W_n)$$ ( W n ) exist. Applications to construction of confidence intervals and simulations are also given.

Suggested Citation

  • Yingqiu Li & Peihan Li & Yushao Wei, 2025. "Cramér Moderate Deviations and Berry-Esseen Bounds for Mandelbrot’s Cascade in a Random Environment," Methodology and Computing in Applied Probability, Springer, vol. 27(2), pages 1-21, June.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:2:d:10.1007_s11009-025-10154-w
    DOI: 10.1007/s11009-025-10154-w
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    References listed on IDEAS

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    2. Grama, Ion & Liu, Quansheng & Miqueu, Eric, 2017. "Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1255-1281.
    3. Doukhan, Paul & Fan, Xiequan & Gao, Zhi-Qiang, 2023. "Cramér moderate deviations for a supercritical Galton–Watson process," Statistics & Probability Letters, Elsevier, vol. 192(C).
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    6. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
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