Estimating large deviation rate functions

TL;DR

This paper demonstrates that even with limited large deviation principles, one can empirically estimate the rate function for random walks using the Attouch-Wets topology, aiding in understanding rare events.

Contribution

It proves that empirical estimates of the rate function are feasible under narrow LDP conditions for Cramer's theorem, especially for heavy-tailed distributions.

Findings

LDP for rate function estimation holds even with narrow LDP conditions

Empirical moment generating function estimates converge in the Attouch-Wets topology

Applicable to heavy-tailed increments in random walks

Abstract

Establishing a Large Deviation Principle (LDP) proves to be a powerful result for a vast number of stochastic models in many application areas of probability theory. The key object of an LDP is the large deviations rate function, from which probabilistic estimates of rare events can be determined. In order make these results empirically applicable, it would be necessary to estimate the rate function from observations. This is the question we address in this article for the best known and most widely used LDP: Cram\'er's theorem for random walks. We establish that even when only a narrow LDP holds for Cram\'er's Theorem, as occurs for heavy-tailed increments, one gets a LDP for estimating the random walk's rate function in the space of convex lower-semicontinuous functions equipped with the Attouch-Wets topology via empirical estimates of the moment generating function. This result may seem surprising as it is saying that for Cram\'er's theorem, one can quickly form non-parametric estimates of the function that governs the likelihood of rare events.