Convergence rate of random geometric sum distributions to the Laplace law

In this paper we modify the Stein method and the auxiliary technique of distributional transformations of random variables. This enables us to estimate the convergence rate of distributions of normalized geometric sums to the Laplace law. For independent summands, the developed approach provides an optimal estimate involving the ideal metric of order 3. New results are also obtained for the Kolmogorov and Kantorovich metrics.