Tail estimates for stochastic fixed point equations via nonlinear renewal theory

Jeffrey F. Collamore and Anand N. Vidyashankar

Stochastic Processes and their Applications, 2013, vol. 123, issue 9, 3378-3429

Abstract: This paper introduces a new approach, based on large deviation theory and nonlinear renewal theory, for analyzing solutions to stochastic fixed point equations of the form V=Df(V), where f(v)=Amax{v,D}+B for a random triplet (A,B,D)∈(0,∞)×R2. Our main result establishes the tail estimate P{V>u}∼Cu−ξ as u→∞, providing a new, explicit probabilistic characterization for the constant C. Our methods rely on a dual change of measure, which we use to analyze the path properties of the forward iterates of the stochastic fixed point equation. To analyze these forward iterates, we establish several new results in the realm of nonlinear renewal theory for these processes. As a consequence of our techniques, we develop a new characterization of the extremal index, as well as a Lundberg-type upper bound for P{V>u}. Finally, we provide an extension of our main result to random Lipschitz maps of the form Vn=fn(Vn−1), where fn=Df and Amax{v,D∗}+B∗≤f(v)≤Amax{v,D}+B.

Keywords: Random recurrence equations; Letac’s principle; Nonlinear renewal theory; Slowly changing functions; Harris recurrent Markov chains; Geometric ergodicity; Large deviations; Cramér–Lundberg theory with stochastic investments; GARCH processes; Extremal index (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (5)

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DOI: 10.1016/j.spa.2013.04.015

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