An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums
By Victor Yurievich Korolev, Irina Shevtsova. 2009
131 citations recorded by the source corpus.
By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F n ,Φ) between the standard normal distribution function Φ and the distribution function F n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β3. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β3+0.429)≤0.33477(1+0.429)β3<0.4784β3 by virtue of the condition β3≥1. The latter inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry–Esseen inequality for Poisson random sums to 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequality. As corollaries, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.
Subfields: Applied Mathematics, Management Science and Operations Research, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/6457c49ee28ad25dfdb46f5fc46ab6f2e3fd50af
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Citations in the corpus
- Self-Normalizing Neural Networks (2017, background): “The Berry-Esseen theorem states that the convergence rate to normality is n − 1 / 2 [18].”
- An improvement of convergence rate estimates in the Lyapunov theorem (2010, methodology): “The proofs of Theorems 1 and 2 are based on a method of Zolotarev improved by using Prawitz’ smoothing inequality [7] and estimates for character� istic functions obtained in [ 2 , 4]. We describe only the main ideas and state the corresponding assertions as lemmas.”
- Shannon Entropy based Randomness Measurement and Test for Image Encryption (2011, background): “ no more than the specified amount. The calculated values of the constant Cwas 7.59 [14] in 1941. Over the years, this value has greatly decreased. So far the best estimate of Cis found as C<0:4784 [17] in 2011, derived from the inequality Eqn. (10). sup z jF Z n (z) (z)j 0:33477(ˆ+ 0:429˙3) ˙3 p ”