A.s. convergence rate for Mandelbrot’s cascade in a random environment

Abstract

. For Mandelbrot’s cascade (Y n ) in an independent and identically distributed (i.i.d.) random environment ξ, we are interested in the a.s. convergence rate of the Mandelbrot’s martingale (W n ) to its limit W, where W n = Y n / E ξ Y n is the normalized partition function. We obtain sufficient conditions under which W−W n has an exponential convergence rate: W − W n = o ( e − n a ) a.s. for some a > 0 explicitly calculated; we also find conditions under which W−W n has a polynomial convergence rate: W − W n = o ( n − α ) a.s. for some α > 0. Similar conclusions hold for Mandelbrot’s cascade in a varying environment.