发表机构
Univ Bretagne Sud, CNRS UMR 6205, LMBA; Universität Hildesheim, Institut für Mathematik und Angewandte Informatik; State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（南布列塔尼大学; 希尔德斯海姆大学; 中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究临界情形下多维仿射随机递归不变测度的尾部行为，证明控制径向集尾部行为的缓变函数有界，并探讨方向性尾部性质。
AI 中文摘要
我们研究了多维仿射随机递归 $V_n = A_n V_{n-1} + B_n$ 的不变Radon测度在无穷远处的行为，其中 $(A_n)_{n \geq 1}$ 是正随机矩阵，$(B_n)_{n \geq 1}$ 是具有非负分量的随机向量，且 $(A_n, B_n)_{n \geq 1}$ 是独立同分布的。在随机矩阵乘积 $A_n \cdots A_1$ 的顶部Lyapunov指数为零的临界情形下，Brofferio、Peigné和Pham [6] 最近建立了具有无限总质量的不变Radon测度的存在性和唯一性（相差一个常数乘法）。他们证明了该测度应用于径向集时的尾部行为由一个缓变函数控制。我们的目标是证明这个缓变函数实际上是有界的。此外，我们还研究了方向性尾部行为。
英文摘要
We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion $V_n = A_n V_{n-1} + B_n,$ where $(A_n)_{n \geq 1}$ are positive random matrices, $(B_n)_{n \geq 1}$ are random vectors with nonnegative entries, and $(A_n, B_n)_{n \geq 1}$ are independent and identically distributed. In the critical regime where the top Lyapunov exponent of the random matrix products $A_n \cdots A_1$ is zero, Brofferio, Peigné and Pham [6] recently established the existence and uniqueness, up to multiplication by a constant, of an invariant Radon measure with infinite total mass. They proved that the tail behavior of this measure when applied to radial sets is governed by a slowly varying function. Our goal is to show that this slowly varying function is actually bounded. Moreover, we investigate directional tail behavior.
Comments25 pages
