On modifications of the Lindeberg and Rotar' conditions in the central limit theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 818-822
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A modification of the Lindeberg and Rotar' conditions was considered in the
papers by Presman and Formanov [Dokl. Math., 99 (2019), pp. 204–207]
and [Dokl. Ross. Akad. Nauk Ser. Mat., 485 (2019), pp. 548–552
(in Russian)].
This modification
was concerned with the sums of absolute (respectively, difference) moments of
order $2+\alpha$ for the distributions of the summands truncated at the unit
level. It was shown that, when checking the normal convergence,
it is sufficient, instead of
checking the convergence to zero of the Lindeberg or Rotar' characteristics
for any $\varepsilon >0$, to check that there exists an
$\alpha >0$ such that a characteristic (introduced in these papers) corresponding
to this $\alpha$ converges to zero. Moreover,
from the existence of such $\alpha$ it follows that the characteristic
corresponding to any $\alpha >0$ also tends to zero. We show
that the moment functions can be changed to more general functions and
describe the class of such functions.
Keywords:
central limit theorem, Lindeberg characteristic, nonclassical version of central limit theorem, Rotar' characteristic, Ibragimov–Osipov–Esseen characteristic.
@article{TVP_2020_65_4_a8,
author = {I. A. Ibragimov and E. L. Presman and Sh. K. Formanov},
title = {On modifications of the {Lindeberg} and {Rotar'} conditions in the central limit theorem},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {818--822},
publisher = {mathdoc},
volume = {65},
number = {4},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a8/}
}
TY - JOUR AU - I. A. Ibragimov AU - E. L. Presman AU - Sh. K. Formanov TI - On modifications of the Lindeberg and Rotar' conditions in the central limit theorem JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2020 SP - 818 EP - 822 VL - 65 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a8/ LA - ru ID - TVP_2020_65_4_a8 ER -
%0 Journal Article %A I. A. Ibragimov %A E. L. Presman %A Sh. K. Formanov %T On modifications of the Lindeberg and Rotar' conditions in the central limit theorem %J Teoriâ veroâtnostej i ee primeneniâ %D 2020 %P 818-822 %V 65 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a8/ %G ru %F TVP_2020_65_4_a8
I. A. Ibragimov; E. L. Presman; Sh. K. Formanov. On modifications of the Lindeberg and Rotar' conditions in the central limit theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 65 (2020) no. 4, pp. 818-822. http://geodesic.mathdoc.fr/item/TVP_2020_65_4_a8/