ASYMPTOTICALLY LINEAR ITERATED FUNCTION SYSTEMS ON THE REAL LINE

• 成果类型: Article
• 署名作者: Alsmeyer, Gerold; Brofferio, Sara; Buraczewski, Dariusz
• 署名单位: University of Munster; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; University of Wroclaw
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1812
• 发表日期: 2023
• 页码: 161-199
• 关键词: RANDOM DIFFERENCE-EQUATIONS; renewal theory; stochastic equation; tail

摘要

Given a sequence of i.i.d. random functions Psi(n) : R -> R, n is an element of 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 is an element of R and n is an element of N. Under the two basic assumptions that the Psi(n) are a.s. continuous at any point in R and asymptotically linear at the endpoints +/-infinity, 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 (Ann. Appl. Probab. (1991) 1 126-166) and can also be viewed as an adaptation of Kesten's work on products of random matrices (Acta Math. (1973) 131 207-248) 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, for example, ARCH models and random logistic transforms.