Estimates for the concentration functions under the weakened moments

V. Yu. Korolev ; A. V. Dorofeyeva

Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 1, pp. 104-121

Résumé

Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the moments of summands of higher orders. The results obtained are extended to Poisson-binomial, binomial, and Poisson random sums. Under the same assumptions, bounds are obtained for the approximation of the concentration functions of mixed Poisson random sums by the corresponding limit distributions. In particular, bounds are put forward for the accuracy of approximation of the concentration functions of geometric, negative binomial, and Sichel random sums by exponential, folded variance gamma, and folded Student distributions. Numerical estimates of all the constants involved are written down explicitly.

Détail

Keywords: distribution function, central limit theorem, folded normal distribution, uniform metric, mixed Poisson random sum, geometric random sum, negative binomial random sum, exponential distribution, folded variance gamma distribution, folded Student distribution
Mots-clés : normal distribution, Poisson-binomial distribution, Poisson-binomial random sum, binomial random sum, Poisson random sum, gamma distribution, inverse gamma distribution, Sichel distribution, Laplace distribution, absolute constant.