Moments for multidimensional Mandelbrot cascades

Abstract

Abstract We consider the distributional equation Z =D ∑k=1NAkZ(k), where N is a random variable taking value in N0 = {0, 1, . . .}, A1, A2, . . . are p x 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, A1, A2, . . .). Let {Yn} 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 = (y1, . . ., yp) ∈ 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 Lα convergence and the αth-moment of the Mandelbrot martingale {Yn} are also established.