On Stein factors for Laplace approximation and their application to random sums

Abstract

In this article, we consider an integral Stein equation for Laplace distribution and solve it using the semigroup approach. Next, we derive regularity estimates for the solution of the Laplace Stein equation. Finally, we apply these estimates to obtain a convergence rate for Laplace approximation of the random geometric sums. We also compare our rate with the existing literature.