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Let (X_n,Y_n) be i.i.d. random vectors. Let W(x) be the partial sum of Y_njust before that of X_n exceeds x>0. Motivated by stochastic models for neuralactivity, 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 oflarge deviations, with a(c,x) a deterministic function and I an open interval.To obtain this uniform exact large deviations principle (LDP), we firstestablish the exponentially fast uniform convergence of a family of renewalmeasures and then apply it to appropriately tilted distributions of X_n and themoment generating function of W(x). The uniform exact LDP is obtained for caseswhere X_n has a subcomponent with a smooth density and Y_n is not a lineartransform of X_n. An extension is also made to the partial sum at the firstexceedance time.

Uniform convergence of exact large deviations for renewal reward processes