Moments for multidimensional mandelbrot cascades

We consider the distributional equation Z^D=Σ^N_κ=1 AκZ(κ), whereN is a random variable taking value in ℕ0 = {0, 1, . . .}, A₁,A₂, . . . 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,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 ϵ (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) ϵ 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 {Y_n} are also established.