广义多维仿射递归的不动点
On fixed points of a generalized multidimensional affine recursion
摘要：
设$G$是一般线性群$\Gl(\mathbb{R}^d)$的一个乘法子半群，该群由正元素矩阵组成，使得每列和每行都包含严格正元素.给定一个$G$值随机矩阵$A$，我们考虑下面的广义多维仿射方程R \stackrel{\mathcal{D}}{=} \sum_{i=1}^N A_iR_i+B，其中$N\ge2$是一个固定的自然数，$A_1,...,A_N$是$A$的独立拷贝，$B\in\mathbb{R}^d$是一个正元素的随机向量，$R_1,...,R_N$是$R\in\mathbb{R}^d$的独立拷贝，它们也有正元素.而且，所有这些都是相互独立的，$\stackrel{\mathcal{D}}{=}$代表分配的平等。借助于Guivarc 'h和Le Page发展的谱理论和Kesten的更新定理，证明了在适当的条件下，存在$χ>0$，使得对每个单位向量$u\in\mathbb{S}^{d-1}$，$¶(\{<R, u>>t\})\asymp t^{-χ},$为$t\to\8$.
英文摘要：
Let $G$ be a multiplicative subsemigroup of the general linear group $\Gl(\mathbb{R}^d)$ which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a $G$--valued random matrix $A$, we consider the following generalized multidimensional affine equation R\stackrel{\mathcal{D}}{=}\sum_{i=1}^N A_iR_i+B, where $N\ge2$ is a fixed natural number, $A_1,...,A_N$ are independent copies of $A$, $B\in\mathbb{R}^d$ is a random vector with positive entries, and $R_1,...,R_N$ are independent copies of $R\in\mathbb{R}^d$, which have also positive entries. Moreover, all of them are mutually independent and $\stackrel{\mathcal{D}}{=}$ stands for the equality in distribution. We will show with the aid of spectral theory developed by Guivarc'h and Le Page and Kesten's renewal theorem, that under appropriate conditions, there exists $χ>0$ such that $¶(\{<R, u>>t\})\asymp t^{-χ},$ as $t\to\8$, for every unit vector $u\in\mathbb{S}^{d-1}$ with positive entries.
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