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