Error bounds in approximations of random sums using gamma-type operators

In this work we deal with approximations of compound distributions, that is, distribution functions of random sums. More specifically, we obtain a discrete compound distribution by replacing each summand in the initial random sum by a discrete random variable whose probability mass function is related to a well-known inversion formula for Laplace transforms [cf. Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. II, second edn. Wiley, New York]. Our aim is to show the advantages that this method has in the context of compound distributions. In particular we give accurate error bounds for the distance between the initial random sum and its approximation when the individual summands are mixtures of gamma distributions.