Tail estimates for stochastic fixed point equations via nonlinear renewal theory

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}+Bf(v)=Amax{v,D}+B for a random triplet (A,B,D)∈(0,∞)×R2(A,B,D)∈(0,∞)×R2. Our main result establishes the tail estimate P{V>u}∼Cu−ξ as u→∞u→∞, providing a new, explicit probabilistic characterization for the constant CC. 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)Vn=fn(Vn−1), where fn=Df and Amax{v,D∗}+B∗≤f(v)≤Amax{v,D}+BAmax{v,D∗}+B∗≤f(v)≤Amax{v,D}+B.