Sharpened upper bounds for the absolute constant in the Berry-Esseen inequality for mixed Poisson random sums

摘要

Let X1, X2, … be independent identically distributed random variables with (1) and let Nλ be a Poisson random variable with parameter λ > 0 independent of the sequence X1, X2, … . We set (for definiteness, we assume that Sλ = 0 for Nλ = 0). Poisson random sums Sλ are popular mathematical models of many reallife objects. In particular, in actuarial mathematics, Sλ is the total insurance premium in the risk classical procedure (in the “dynamical” case) or that corresponding to a certain portfolio (in the “static” case); see, e.g., [1]. Numerous examples of applied problems from diverse areas in which Poisson random sums arise are given in [9, 7].