Geodesic
Convergence rate estimates in the global CLT for compound mixed Poisson distributions
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 89-116

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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.
Mots-clés : Poisson random sum, Laplace distribution
Keywords: geometric random sum, compound Poisson distribution, central limit theorem (CLT), convergence rate estimate, normal approximation, Berry–Esseen inequality, asymptotically exact constant. 
@article{TVP_2018_63_1_a4,
     author = {I. G. Shevtsova},
     title = {Convergence rate estimates in the global {CLT} for compound mixed {Poisson} distributions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {89--116},
     publisher = {mathdoc},
     volume = {63},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a4/}
}
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I. G. Shevtsova. Convergence rate estimates in the global CLT for compound mixed Poisson distributions. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 89-116. http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a4/