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

Yingqiu Li (), Peihan Li () and Yushao Wei ()
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Yingqiu Li: Changsha University of Science and Technology
Peihan Li: Changsha University of Science and Technology
Yushao Wei: Changsha University of Science and Technology

Methodology and Computing in Applied Probability, 2025, vol. 27, issue 2, 1-21

Abstract: 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.

Keywords: Mandelbrot’s cascade; Random environment; Cramér moderate deviations; Berry-Esseen bounds; Harmonic moments; 60J80; 60K37; 60F10 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s11009-025-10154-w

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