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).
This is a preview of subscription content, log in to check access.
R. A. Gabdullin, V. A. Makarenko, and I. G. Shevtsova, “Esseen–Rozovskii type estimates for the rate of convergence in the Lindeberg theorem,” J. Math. Sci., 234, No. 6, 847–885 (2018).MathSciNetCrossRefGoogle Scholar
R. A. Gabdullin, V. A. Makarenko, and I. G. Shevtsova, “A generalization of the Wang–Ahmad inequality,” J. Math. Sci., 237, No. 5, 646–651 (2019).CrossRefGoogle Scholar
C.-G. Esseen, “On the Liapounoff limit of error in the theory of probability,” Ark. Mat. Astron. Fys., A28, No. 9, 1–19 (1942).MathSciNetzbMATHGoogle Scholar
V. Korolev and A. Dorofeyeva, “Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions,” Lith. Math. J., 57, No. 1, 38–58 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
V. Yu. Korolev and S. V. Popov, “Improvement of convergence rate estimates in the central limit theorem under weakened moment conditions,” Dokl. Math., 86, No. 1, 506–511 (2012).MathSciNetCrossRefzbMATHGoogle Scholar