Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations

Abstract

Given $d\ge 1$, let $(A_i{)}_{i\ge 1}$ be a sequence of random $d\times d$ real matrices and $Q$ be a random vector in ${\mathbb{R}}^d$. We consider fixed points of multivariate smoothing transforms, i.e. random variables $X\in {\mathbb{R}}^d$ satisfying $X$ has the same law as ${\sum }_{i\ge 1}{A}_i{X}_i+Q$, where $(X_i{)}_{i\ge 1}$ are i.i.d. copies of $X$ and independent of $(Q,(A_i{)}_{i\ge 1})$. The existence of fixed points that can attract point masses can be shown by means of contraction arguments. Let $X$ be such a fixed point. Assuming that the action of the matrices is expanding as well with positive probability, it was shown in a number of papers that there is $\beta >0$ with ${\lim }_{t\to \infty }{t}^{\beta }\mathbb{P}(<u,X>>t)=K\cdot f(u)$, where $u$ denotes an arbitrary element of the unit sphere and $f$ a positive function and $K\ge 0$. However in many cases it was not established that $K$ is indeed positive. In this paper, under quite general assumptions, we prove that ${\mathrm{liminf}}_{t\to \infty }{t}^{\beta }\mathbb{P}(<u,X>>t)>0,$ completing, in particular, the results of arXiv:1111.1756 and arXiv:1206.1709.

17 pages

References in corpus (4)

• Heavy tailed solutions of multivariate smoothing transforms
• Fixed points of multivariate smoothing transforms with scalar weights
• Precise tail asymptotics of fixed points of the smoothing transform with general weights
• Ranking algorithms on directed configuration networks