Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations

D. Buraczewski, S. Mentemeier

2017, v.23, №1, 147-170

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 $\R^d$. We
consider fixed points of multivariate smoothing transforms, i.e.\ random variables $X\in \Rd$ satisfying
$$
X \text{ has the same law as } \sum_{i \ge 1} A_i X_i + Q,
$$
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.
%
%By means of contraction arguments, the existence of fixed points, which attract point masses, can be %shown.
Let $X$ be such a fixed point. Assuming that the action of the matrices is expanding as well as contracting with positive probability, it was shown in a number of papers that there is $\beta >0$ with
$\lim_{t \to \infty} t^\beta \P{\skalar{u,X}>t} = K f(u)$, where
$u$ denotes an arbitrary element of the unit sphere, $f$ is 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
$$
\liminf_{t\to\8} t^{\beta} \P{\langle u,X \rangle > t}> 0,
$$
completing, in particular, the results of \cite{Mirek2013} and \cite{BDMM2013}.

Keywords: smoothing transform, heavy tails, products of random matrices, branching

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