On Modifications of the Lindeberg and Rotar' Conditions in the Central Limit Theorem

Ибрагимов, И. А.; Presman, E. L.; Formanov, Sh. K.

doi:10.1137/s0040585x97t990186

Cited by 6 publications

(2 citation statements)

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“…Further research might include construction of the estimates of the rate of convergence in non-classical Lindeberg's theorem whose conditions were obtained in a recent paper [30] or some extensions to the case of dependent random summands (see [31]).…”

Section: Discussionmentioning

confidence: 99%

Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

2021

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Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones.

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Section: Discussionmentioning

confidence: 99%

Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

2021

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“…Furthermore, in [22] he found an existence criterion for Chebyshev-Cramér (Chebyshev-Edgeworth) expansions, and in the quite recent join paper with E. L. Presman and Sh. K. Formanov [73] they presented conditions for the validity of the central limit theorem, which are equivalent to Lindeberg's and Rotar's ones and in which the moment functions can be replaced by more general functions.…”

Section: Mathematical Lifementioning

confidence: 99%