Convergence rate estimates in the global CLT for compound mixed Poisson distributions

Using the estimates of the accuracy of the normal approximation to distributions of Poisson-binomial random sums from [I. G. Shevtsova, Theory Probab. Appl., 62 (2018), pp. 278–294], we obtain moment-type estimates of the rate of convergence in the central limit theorem for Poisson and mixed Poisson random sums in the uniform and mean metrics. As corollaries, we provide estimates of the accuracy of the approximation to distributions of negative binomial random sums by the normal law (with the growth of the shape parameter) and by the variance-gamma mixture of the normal law (as the “success probability” tends to zero); in particular, we present estimates of the accuracy of the Laplace approximation to distributions of geometric random sums.