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This paper presents precise large deviation estimates for solutions tostochastic 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, weprovide 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 upperbound of the form P(V > u) <= D(u) u^-r. To this end, we introduce a novel dualchange of measure on a random time interval and analyze the path properties,using nonlinear renewal theory, of the Markov chain resulting from the forwarditeration of the given stochastic fixed point equation. In the process, weestablish several new results in the realm of nonlinear renewal theory forthese processes. As a consequence of our techniques, we also establish a newcharacterization of the extremal index. Finally, we provide some extensions ofour methods to Markov-driven sequences.

Tail estimates for stochastic fixed point equations via nonlinear renewal theory