Moments for multidimensional Mandelbrot cascades

DOI10.1017/JPR.2015.4zbMATH Open1341.60027arXiv1405.2681OpenAlexW2295295994MaRDI QIDQ2804408

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Publication date: 29 April 2016

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Abstract: We consider the distributional equation

e x t b f Z s t a c k r e l d = s u m_{k = 1}^{N} e x t b f A_{k} e x t b f Z (k)

, where

N

is a random variable taking value in

m a t h b b N_{0} = 0, 1, c d o t s

,

e x t b f A_{1}, e x t b f A_{2}, c d o t s

are

p i m e s p

non-negative random matrix, and

e x t b f Z, e x t b f Z (1), e x t b f Z (2), c d o t s

are

i . i . d

random vectors in in

m a t h b b R_{+}^{p}

with

m a t h b b R_{+} = [0, i n f t y)

, which are independent of

(N, e x t b f A_{1}, e x t b f A_{2}, c d o t s)

. Let

m a t h b f Y_{n}

be the multi-dimensional Mandelbrot's martingale defined as sums of products of random matrixes indexed by nodes of a Galton-Watson tree plus an appropriate vector. Its limit

m a t h b f Y

is a solution of the equation above. For

a l p h a > 1

, we show respectively a sufficient condition and a necessary condition for

m a t h b b E | m a t h b f Y |^{a} l p h a i n (0, i n f t y)

. Then for a non-degenerate solution

m a t h b f Z

of the equation above, we show the decay rates of

m a t h b b E e^{- m a t h b f t c d o t m a t h b f Z}

as

| m a t h b f t | i g h t a r r o w i n f t y

and those of the tail probability

m a t h b b P (m a t h b f y c d o t m a t h b f Z l e q x)

as

x i g h t a r r o w 0

for given

m a t h b f y = (y^{1}, c d o t s, y^{p}) i n m a t h b b R_{+}^{p}

, and the existence of the harmonic moments of

m a t h b f y c d o t m a t h b f Z

. As application, these above results about the moments (of positive and negative orders) of

m a t h b f Y

are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrixes of the equation above are complex, a sufficient condition for the

L^{a} l p h a

convergence and the

a l p h a

th-moment of the Mandelbrot's martingale

m a t h b f Y_{n}

is also established.

Full work available at URL: https://arxiv.org/abs/1405.2681

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