Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations

TL;DR

This paper establishes precise tail asymptotics for fixed points of multivariate smoothing transformations, proving that the tail probability decays polynomially with a positive constant, under general conditions.

Contribution

It proves the positivity of the tail constant for multivariate fixed points, completing previous results and providing a comprehensive understanding of tail behavior.

Findings

Tail probability decays as a power law with exponent β

The tail constant K is shown to be positive under general conditions

Completes previous partial results on tail asymptotics

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 $\liminf_{t\to\infty} t^{\beta} \mathbb{P} (<u,X >> t)> 0,$ completing, in particular, the results of arXiv:1111.1756 and arXiv:1206.1709.