Bounds for the concentration functions of random sums under relaxed moment conditions

2016/08/09 by V. Yu. Korolev, Korolev, V. Yu., A. V. Dorofeeva +1
Mathematics · #FOS: Mathematics #Mathematical functions and polynomials #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1608.03001

openalex publication_date 2016/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 obtained results are extended to Poisson-binomial, binomial and Poisson random sums. Under the same assumptions, the bounds are obtained for the approximation of the concentration functions of mixed Poisson random sums by the corresponding limit distributions. In particular, bounds are obtained for the accuracy of approximation of the concentration functions of geometric, negative binomial and Sichel random sums by the exponential, the folded variance gamma and the folded Student distribution. Numerical estimates of all the constants involved are written out explicitly.

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