Discounted payments theorems for large deviations

Abstract

Let Z(t) = Σ _j=1^N(t) X_j, t ≥ 0, be a stochastic process, where X_j are independent identically distributed random variables, and N(t) is non-negative integer-valued process with independent increments. Throughout, we assume that N(t) and X_j are independent. The paper considers normal approximation to the distribution of properly centered and normed random variable Z_δ =∫₀^∞e^{- δt} dZ(t), δ > 0, taking into consideration large deviations both in the Cramér zone and the power Linnik zones. Also, we obtain a nonuniform estimate in the Berry–Essen inequality.