Asymptotically linear iterated function systems on the real line

TL;DR

This paper studies the tail behavior of stationary distributions in asymptotically linear iterated function systems on the real line, extending renewal theory and applying to various fields like finance and population dynamics.

Contribution

It extends Goldie's implicit renewal theory to asymptotically linear systems and adapts Kesten's work to one-dimensional function systems.

Findings

Provides tail asymptotics for stationary laws

Extends renewal theory to new class of systems

Applicable to finance, queuing, and population models

Abstract

Given a sequence of i.i.d. random functions $\Psi_{n}:\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$, we consider the iterated function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$ and $X_{n}^{x}:=\Psi_{n-1}(X_{n-1}^{x})$ for $x\in\mathbb{R}$ and $n\in\mathbb{N}$. Under the two basic assumptions that the $\Psi_{n}$ 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.