Uniform convergence of exact large deviations for renewal reward processes (Q2456051)

Let \((X_n , Y_n)\) be i.i.d. random variables in \(\mathbb{R}^2\). Denote \( S_n = \sum_1^n X_k , \;N(x) = \max \{ n : \max_{1\leq k\leq n} S_k \leq x \}\) and \(W(x) = \sum_1^{N(x)} Y_k \). In this article the author proves the uniform exact large deviations principle for the random sum \(W(x)\), namely \[ \sup_{t\in J} | a(t,x)P( W(x)\geq tx) -1 | =o(1), \quad x\rightarrow \infty , \] (with \(J\) being an open interval and \(a(t,x)\) a certain deterministic function) for cases where \(X_n\) has a subcomponent with a smooth density and \(Y_n\) is not a linear transform of \(X_n\).