On multidimensional Mandelbrot's cascades

Dariusz Buraczewski (1) , Ewa Damek (1) , Yves Guivarc'H (2)

1 Instytut Matematyczny
2 IRMAR - Institut de Recherche Mathématique de Rennes

Let $Z$ be a random variable with values in a proper closed convex cone $C\subset \R^d$, $A$ a random endomorphism of $C$ and $N$ a random integer. Given $N$ independent copies $(A_i,Z_i)$ of $(A,Z)$ we define a new random variable $\wh Z = \sum_{i=1}^N A_i Z_i$. Let $T$ be the corresponding transformation on the set of probability measures on $C$. We study existence and properties of fixed points of $T$. Previous one dimensional results on existence of fixed points of T as well as on homogeneity of their tails are extended to higher dimensions.

Mots clés: Mandelbrot's cascade; products of random matrices; renewal theorem; fixed points; asymptotic behavior

Domaines: Probabilités [math.PR]

Dariusz Buraczewski, Ewa Damek, Yves Guivarc'H. On multidimensional Mandelbrot's cascades. Journal of Difference Equations and Applications, 2014, 20 (11), pp.1523-1567. ⟨hal-00769633⟩