Informatics and Applications
2021, Volume 15, Issue 1, pp 116-121
ON THE ACCURACY OF THE NORMAL APPROXIMATION UNDER THE VIOLATION OF THE NORMAL CONVERGENCE
- V. Yu. Korolev
- A. V. Dorofeeva
Abstract
When solving applied problems in various fields, it is conventional to use the normal approximation to the distribution of data with additive structure. As a criterion of the adequacy of such a model, it is possible to use bounds for the convergence rate in the central limit theorem (CLT) of the probability theory stating that under certain conditions (say, under the Lindeberg condition), the total effect of very many random factors acts as a random variable with the normal distribution. The classical bounds for the convergence rate in the CLT such as the Berry-Esseen inequality are proved under the condition that the third moments of the summands exist. Also, bounds are known that require the existence of the moments of orders 2 + 5 with 0 < 5 < 1. If only the moments of the second order exist, then the convergence in the CLT can be arbitrarily slow. But if the moments of the summands of the second order do not exist, then the convergence of the distributions of sums of independent random variables to the normal law does not take place. It is practically impossible to reliably check the conditions of the central limit theorem with the limited size of the available sample. Therefore, the question of what is the real accuracy of the normal approximation if it is theoretically impossible is of great interest. Moreover, in some situations, in computer simulation of sums of random variables whose distributions belong to the domain of attraction of the stable distribution with the characteristic exponent less than two, as the number of summands grows, first, the distance between the distribution of the normalized sum and the normal law decreases and starts to increase only when the number of summands becomes sufficiently large. In this paper, an attempt is undertaken to give some theoretical explanation of this effect and to give an answer to the question posed above.
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[+] About this article
Title
ON THE ACCURACY OF THE NORMAL APPROXIMATION UNDER THE VIOLATION OF THE NORMAL CONVERGENCE
Journal
Informatics and Applications
2021, Volume 15, Issue 1, pp 116-121
Cover Date
2021-03-30
DOI
10.14357/19922264210116
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
central limit theorem; accuracy of normal approximation; heavy tails; uniform distance
Authors
V. Yu. Korolev  ,  and A. V. Dorofeeva
Author Affiliations
Faculty of Computational Mathematics and Cybernetics, M.V. Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow 119991, Russian Federation
Institute of Informatics Problems, Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
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