Moments for multidimensional Mandelbrot cascades

Abstract

We consider the distributional equationwhere N is a random variable taking value in N 0 = {0, 1, . . .}, A 1 , A 2 , . . . are p × p nonnegative random matrices, and Z, Z(1), Z(2), . . . , are independent and identically distributed random vectors in R p + with R + = [0, ∞), which are independent of (N, A 1 , A 2 , . . .). Let {Y n } be the multidimensional Mandelbrot martingale defined as sums of products of random matrices indexed by nodes of a Galton-Watson tree plus an appropriate vector. Its limit Y is a solution of the equation above. For α > 1, we show a sufficient condition for E Y α ∈ (0, ∞). Then for a nondegenerate solution Z of the distributional equation above, we show the decay rates of Ee −t·Z as t → ∞ and those of the tail probability P(y ·Z ≤ x) as x → 0 for given y = (y 1 , . . . , y p ) ∈ R p + , and the existence of the harmonic moments of y ·Z. As an application, these results concerning the moments (of positive and negative orders) of Y are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrices of the equation above are complex, a sufficient condition for the L α convergence and the αth-moment of the Mandelbrot martingale {Y n } are also established.