多元光滑变换吸引不动点的精确尾渐近性
Precise Tail Asymptotics for Attracting Fixed Points of Multivariate Smoothing Transformations
摘要：
给定$d \ge 1$，设$(A_i)_{i\ge 1}$是随机$d\times d$实矩阵序列，$Q$是$\mathbb{R}^d$中的随机向量。我们考虑多元平滑变换的不动点，即随机变量$X\in \mathbb{R}^d$满足$X$与$\sum_{i \ge 1} A_i X_i + Q$具有相同的规律，其中$(X_i)_{i \ge 1}$是独立同分布的。$X$的拷贝，并且独立于$(Q, (A_i)_{i \ge 1})$。可以吸引点质量的不动点的存在可以通过收缩论证来证明。让$X$成为这样一个固定点。假设矩阵的作用也以正概率扩展，许多论文都表明存在带有$\lim_{t \to \infty} t^β\mathbb{P}(<u,X > >t ) = K\cdot f(u)$的$β>0$，其中$u$表示单位球面的任意元素，$f$表示正函数，$K \ge 0$表示正函数。然而，在许多情况下，并不确定$K$确实是阳性的。在本文中，在相当一般的假设下，我们证明了$\liminf_{t\to\infty} t^β \mathbb{P} (<u,X >> t)> 0,$完成，特别是arXiv：1111.1756和arXiv：1206.1709的结果。
英文摘要：
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 $β>0$ with $\lim_{t \to \infty} t^β\mathbb{P}(<u,X > >t ) = K\cdot f(u)$, 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 $\liminf_{t\to\infty} t^β \mathbb{P} (<u,X >> t)> 0,$ completing, in particular, the results of arXiv:1111.1756 and arXiv:1206.1709.
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