Cramér moderate deviations and Berry-Esseen bounds for Mandelbrot’s cascade in a random environment

Abstract

Let (Y n) be a Mandelbrot’s cascade in an independent and identically distributed (i.i.d.) random environment ξ. According to the existence of the annealed Laplace transform of the limit variable W = lim n→∞ W n , where 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 under the annealed law P are established. It is also shown that harmonic moments of Mandelbrot’s martingale (W n) exist. 2010 MSC: 60J80, 60K37, 60F10