Abstract
Given a sequence of i.i.d. random functions , , we consider the iterated function system and Markov chain, which is recursively defined by and for and . Under the two basic assumptions that the are a.s. continuous at any point in and asymptotically linear at the “endpoints” , 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, models and random logistic transforms.
Funding Statement
The first author was partially funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.
The third author was partially supported by the National Science Center, Poland (Grant No. 2019/33/B/ST1/00207).
Acknowledgments
The authors would like to express their sincere gratitude to an anonymous referee whose numerous suggestions and constructive comments helped to improve the final version of this article.
Citation
Gerold Alsmeyer. Sara Brofferio. Dariusz Buraczewski. "Asymptotically linear iterated function systems on the real line." Ann. Appl. Probab. 33 (1) 161 - 199, February 2023. https://doi.org/10.1214/22-AAP1812
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