Tail estimates for stochastic fixed point equations via nonlinear renewal theory

TL;DR

This paper derives precise tail estimates for solutions to stochastic fixed point equations using nonlinear renewal theory, introducing new techniques and characterizations for extremal behavior.

Contribution

It provides explicit tail asymptotics and a novel dual change of measure approach for stochastic fixed point equations, advancing nonlinear renewal theory.

Findings

Explicit tail asymptotics for V

A new dual change of measure technique

Characterization of the extremal index

Abstract

This paper presents precise large deviation estimates for solutions to stochastic fixed point equations of the type V =_d f(V), where f(v) = Av + g(v) for a random function g(v) = o(v) a.s. as v tends to infinity. Specifically, we provide an explicit characterization of the pair (C,r) in the tail estimate P(V > u) ~ C u^-r as u tends to infinity, and also present a Lundberg-type upper bound of the form P(V > u) <= D(u) u^-r. To this end, we introduce a novel dual change of measure on a random time interval and analyze the path properties, using nonlinear renewal theory, of the Markov chain resulting from the forward iteration of the given stochastic fixed point equation. In the process, we establish several new results in the realm of nonlinear renewal theory for these processes. As a consequence of our techniques, we also establish a new characterization of the extremal index. Finally, we provide some extensions of our methods to Markov-driven sequences.