A Generalization of the Rozovskii Inequality

Adopting ideas of Katz (1963), Petrov (1965), Wang and Ahmad (2016), and Gabdullin, Makarenko, and Shevtsova (2016), we generalize the Rozovskii inequality (1974) which provides an estimate of the accuracy of the normal approximation to distribution of a sum of independent random variables in terms...

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Published inJournal of mathematical sciences (New York, N.Y.) Vol. 237; no. 6; pp. 775 - 781
Main Authors Gabdullin, R. A., Makarenko, V.A., Shevtsova, I. G.
Format Journal Article
LanguageEnglish
Published New York Springer US 09.03.2019
Springer
Springer Nature B.V
Subjects
Equality
Independent variables
Inequality
Mathematics
Mathematics and Statistics
Random variables
Weighting functions
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Summary:Adopting ideas of Katz (1963), Petrov (1965), Wang and Ahmad (2016), and Gabdullin, Makarenko, and Shevtsova (2016), we generalize the Rozovskii inequality (1974) which provides an estimate of the accuracy of the normal approximation to distribution of a sum of independent random variables in terms of the absolute value of the sum of truncated in a fixed point third-order moments and the sum of the second-order tails of random summands. The generalization is due to introduction of a truncation parameter and a weighting function from a set of functions originally introduced by Katz (1963). The obtained inequality does not assume finiteness of moments of random summands of order higher than the second and may be even sharper than the celebrated inequalities of Berry (1941), Esseen (1942, 1969), Katz (1963), Petrov (1965), and Wang & Ahmad (2016).
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-019-04203-2