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We improve bounds of accuracy of the normal approximation to the distribution of a sum of independent random variables under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. We extend these results to Poisson binomial, binomial, and Poisson random sums. Under the same conditions, we obtain bounds for the accuracy of approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, we construct these bounds for the accuracy of approximation of the distributions of geometric, negative binomial, and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma, and Student distributions, respectively.

Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions∗