Uniform convergence of exact large deviations for renewal reward processes

TL;DR

This paper proves a uniform convergence result for large deviation probabilities in renewal reward processes, extending classical large deviations theory with applications to neural activity models.

Contribution

It establishes a uniform exact large deviations principle for renewal reward processes with specific conditions on the distributions, including cases with smooth densities.

Findings

Uniform convergence of large deviation probabilities is proven.

The results apply to renewal reward processes with certain distributional assumptions.

An extension to partial sums at the first exceedance time is provided.

Abstract

Let (X_n,Y_n) be i.i.d. random vectors. Let W(x) be the partial sum of Y_n just before that of X_n exceeds x>0. Motivated by stochastic models for neural activity, uniform convergence of the form $\sup_{c\in I}|a(c,x)\operatorname {Pr}\{W(x)\gecx\}-1|=o(1)$, $x\to\infty$, is established for probabilities of large deviations, with a(c,x) a deterministic function and I an open interval. To obtain this uniform exact large deviations principle (LDP), we first establish the exponentially fast uniform convergence of a family of renewal measures and then apply it to appropriately tilted distributions of X_n and the moment generating function of W(x). The uniform exact LDP is obtained for cases where X_n has a subcomponent with a smooth density and Y_n is not a linear transform of X_n. An extension is also made to the partial sum at the first exceedance time.