On multidimensional Mandelbrot's cascades

Abstract

Let $Z$ be a random variable with values in a proper closed convex cone $C\subset {\mathbb{R}}^d$, $A$ a random endomorphism of $C$ and $N$ a random integer. We assume that $Z$, $A$, $N$ are independent. Given $N$ independent copies $(A_i,Z_i)$ of $(A,Z)$ we define a new random variable $\hat{Z}={\sum }_{i=1}^N{A}_i{Z}_i$. Let $T$ be the corresponding transformation on the set of probability measures on $C$ i.e. $T$ maps the law of $Z$ to the law of $\hat{Z}$. If the matrix $\mathbb{E}[N]\mathbb{E}[A]$ has dominant eigenvalue 1, we study existence and properties of fixed points of $T$ having finite nonzero expectation. Existing one dimensional results concerning $T$ are extended to higher dimensions. In particular we give conditions under which such fixed points of $T$ have multidimensional regular variation in the sense of extreme value theory and we determine the index of regular variation.

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