On multidimensional Mandelbrot cascades

Abstract

Let Z be a random variable with values in a proper closed convex cone C ⊂ R d , A a random endomorphism of C and N a random integer. We assume that Z, A, N are independent.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 Z. If the matrix E[N ]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.

On fixed points of a generalized multidimensional affine recursion

“…(1.1) with B = 0, was studied by Liu in a series of articles (see for instance [18] and the references given there). We are also motivated by the recent results of Buraczewski et al [3], where the authors considered the multidimensional version of Eq. (1.1) with B = 0, and established formula (1.2) with the help of Kesten's renewal theorem [17] and the spectral method developed by Guivarc'h and Le Page [6,7].…”

Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions

“…If T is not irreducible, we can consider the subspace V + (T ) generated by the dominant eigenvectors of the elements of T prox ; then V + (T ) is T -invariant (see [21] p 120) and, if T + is the restriction of T to V + (T ), then T + satisfies condition i-p. In that way our results below could be used in reducible situations (see for example [9]). Definition 2.4.…”

“…This property plays an essential role in the renewal theorems of section 4 as well as in section 5 for d > 1. In the context of non negative matrices a modified form is also valid (see [9]); in ( [35]), Theorem A) under weaker conditions on T , its conclusion is assumed as an hypothesis. If d = 1, we will need to assume it, i.e we will assume that T is non-arithmetic ; if T = [suppµ] satisfies this condition, we say that µ is non-arithmetic.…”

“…The arguments developed in the proof of homogeneity at infinity for ρ can also be used in the study of certain quasi-linear equations which occur in various domains related to branching random walks in particular (see [20]), [42]). For example the description of the shape at infinity of the fixed points of the multidimensional version of the "smoothing transformation" considered in [13] in the context of infinite particle systems in interaction depends on such arguments (see [9]). In an econometrical context, the stochastic recursion (R) can be interpreted as a mechanism which, in the long run, produces debt or wealth accumulation with a specific homogeneous structure at large values; hence,in the natural setting of affine stochastic recursions, this mechanism "explains" the remarkable power law asymptotic shape of wealth distribution empirically discovered by the economist V. Pareto ([43]).…”

Fractal generation method based on asymptote family of generalized Mandelbrot set and its application

“…Buraczewski et al studied Mandelbrot set cascades in multidimensions [5]. Ashish et al researched features of Mandelbrot set in Noor orbit [3].…”