March 2016 Moments for multidimensional Mandelbrot cascades
Chunmao Huang
Author Affiliations +
J. Appl. Probab. 53(1): 1-21 (March 2016).


We consider the distributional equation Z =Dk=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-tZ as |t| → ∞ and those of the tail probability P(yZx) as x → 0 for given y = (y1, . . ., yp) ∈ R+p, and the existence of the harmonic moments of yZ. 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.


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Chunmao Huang. "Moments for multidimensional Mandelbrot cascades." J. Appl. Probab. 53 (1) 1 - 21, March 2016.


Published: March 2016
First available in Project Euclid: 8 March 2016

zbMATH: 1341.60027
MathSciNet: MR3471942

Primary: 60J80
Secondary: 60G42

Keywords: harmonic moments , Mandelbrot martingales , moments , multibranching random walks , multiplicative cascades

Rights: Copyright © 2016 Applied Probability Trust


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Vol.53 • No. 1 • March 2016
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