On Stein factors for Laplace approximation and their application torandom sums

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