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Teoriya Veroyatnostei i ee Primeneniya, 2017, Volume 62, Issue 1, Pages 122–144
DOI: https://doi.org/10.4213/tvp5091
(Mi tvp5091)
 

On bounds for characteristic functions of the powers of asymptotically normal random variables

Yu. V. Prokhorova, F. Götzeb, V. V. Ulyanovcd

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Bielefeld University, Department of Mathematics
c Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics
d National Research University "Higher School of Economics" (HSE), Moscow
References:
Abstract: We obtain upper bounds for the absolute values of the characteristic functions of the $k$th powers of asymptotically normal random variables. Estimates are proved for the case when asymptotically normal random variables are normalized sums of independent identically distributed summands with a “regular” distribution. Possible generalizations are considered. The estimates extend the results of previous studies, where for the distributions of the summands, the presence of either a discrete or an absolutely continuous component was required. The proofs of the bounds are based on the stochastic generalization of the I. M. Vinogradov mean value theorem, which is also obtained in the present paper.
Keywords: powers of random variables, bounds for characteristic functions, the Vinogradov mean value theorem, stochastic generalization.
Funding agency Grant number
Russian Science Foundation 14-11-00196
This work was supported by the project of the Russian Science Foundation 14-11-00196, performed at Lomonosov Moscow State University.
Received: 05.09.2016
Accepted: 20.10.2016
English version:
Theory of Probability and its Applications, 2018, Volume 62, Issue 1, Pages 98–116
DOI: https://doi.org/10.1137/S0040585X97T988514
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: Yu. V. Prokhorov, F. Götze, V. V. Ulyanov, “On bounds for characteristic functions of the powers of asymptotically normal random variables”, Teor. Veroyatnost. i Primenen., 62:1 (2017), 122–144; Theory Probab. Appl., 62:1 (2018), 98–116
Citation in format AMSBIB
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