Uniform convergence of exact large deviations for renewal reward processes

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∈I|a(c, x)Pr {W(x)≥cx}−1|=o(1), x→∞, 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.