Moments for multidimensional Mandelbrot cascades

Abstract

We consider the distributional equation Z =^D ∑_k=1^NA_kZ(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 {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 = (y¹, . . ., 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.