arXiv 2102.022992021-02math.PR

实直线上的渐近线性迭代函数系

Asymptotically linear iterated function systems on the real line

Gerold Alsmeyer, Sara Brofferio, Dariusz Buraczewski

备注：: 49 pages, 3 figures

摘要：

给定一个序列的i.i.d.随机函数$Ψ_{n}:\mathbb{R}\to\mathbb{R}$，$n\in\mathbb{N}$，我们考虑了$X_{0}^{x}:=x$和$X_{n}^{x}:=Ψ_{n-1}(X_{n-1}^{x})$递归定义的迭代函数系统和马尔可夫链，对于$x\in\mathbb{R}$和$n\in\mathbb{N}$ .在$Ψ_{n}$是a.s.在$\mathbb{R}$中任意点连续，在“端点”$\pm\infty$渐近线性的条件下，利用马尔可夫更新理论研究了这类马尔可夫链平稳律的尾部行为.我们的方法提供了一个扩展戈尔迪的隐式更新理论，也可以被视为适应凯斯滕的工作产品的随机矩阵一维函数系统所描述的。我们的结果有应用在不同领域的应用概率，如排队论，计量经济学，金融数学和人口动力学。我们的结果有应用在不同领域的应用概率，如排队论，计量经济学，金融数学和人口动力学，例如，随机模型和随机逻辑变换。

英文摘要：

Given a sequence of i.i.d. random functions $Ψ_{n}:\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$, we consider the iterated function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$ and $X_{n}^{x}:=Ψ_{n-1}(X_{n-1}^{x})$ for $x\in\mathbb{R}$ and $n\in\mathbb{N}$. Under the two basic assumptions that the $Ψ_{n}$ are a.s. continuous at any point in $\mathbb{R}$ and asymptotically linear at the "endpoints" $\pm\infty$, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie's implicit renewal theory and can also be viewed as an adaptation of Kesten's work on products of random matrices to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, e.g. ARCH models and random logistic transforms.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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