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- W1541080589 abstract "Given $d ge 1$, let $(A_i)_{ige 1}$ be a sequence of random $dtimes d$ real matrices and $Q$ be a random vector in $mathbb{R}^d$. We consider fixed points of multivariate smoothing transforms, i.e. random variables $Xin mathbb{R}^d$ 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 $lim_{t to infty} t^beta mathbb{P}( >t ) = Kcdot f(u)$, 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 $liminf_{ttoinfty} t^{beta} mathbb{P} ( > t)> 0,$ completing, in particular, the results of arXiv:1111.1756 and arXiv:1206.1709." @default.
- W1541080589 created "2016-06-24" @default.
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- W1541080589 date "2015-02-09" @default.
- W1541080589 modified "2023-09-25" @default.
- W1541080589 title "Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations" @default.
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