On multidimensional Mandelbrot's cascades

TL;DR

This paper extends the theory of Mandelbrot's cascades to multiple dimensions, analyzing fixed points of a transformation on probability measures in convex cones, and characterizing their regular variation properties.

Contribution

It generalizes one-dimensional results to higher dimensions, providing conditions for fixed points with finite expectation and their regular variation characteristics.

Findings

Fixed points exist under certain eigenvalue conditions.

Fixed points exhibit multidimensional regular variation.

The index of regular variation is explicitly determined.

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.