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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1967, Volume 12, Issue 3, Pages 506–519 (Mi tvp731)

On the Chebyshev–Cramér asymptotic expansions

I. A. Ibragimov

Abstract: Let $\xi_1,…,\xi_n,…$ be a sequence of independent identically distributed random variables with a distribution function (d.f.) $F(x)$ and let $\mathbf E\xi_i=0$, $\mathbf D\xi_i=1$. Denote $\mathbf P\{\frac1{\sqrt n}\sum_1^n\xi_i<x\}=F_n(x)$. Let $\beta_1,\beta_2,…,\beta_n,…$ be a numerical sequence such that $\beta_1=\mathbf E\xi_1=0$, $\beta_2=\mathbf E\xi_1^2=1$ and the other $\beta_s$ are arbitrary. Let us connect with the $\beta$-sequence the sequence $\{Q_n(x)\}$ of the Chebyshev–Cramér polynomials constructed in such a way as if $\{\beta_n\}$ were the sequence of moments of some distribution. We investigate the rate of convergence of the difference
$$\sup\limits_n|F_n(x)-[\Phi(x)+\frac1{\sqrt{2\pi}}e^{-x^2/2}\sum_{s=1}^k\frac{Q_s(x)}{n^{s/2}}]|$$
to zero (here $\Phi(x)$ is the normal d.f.).

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English version:
Theory of Probability and its Applications, 1967, 12:3, 455–469

Bibliographic databases:

Citation: I. A. Ibragimov, “On the Chebyshev–Cramér asymptotic expansions”, Teor. Veroyatnost. i Primenen., 12:3 (1967), 506–519; Theory Probab. Appl., 12:3 (1967), 455–469

Citation in format AMSBIB
\Bibitem{Ibr67}
\by I.~A.~Ibragimov
\paper On the Chebyshev--Cram\'er asymptotic expansions
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 3
\pages 506--519
\mathnet{http://mi.mathnet.ru/tvp731}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=216550}
\zmath{https://zbmath.org/?q=an:0201.51001}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 3
\pages 455--469
\crossref{https://doi.org/10.1137/1112055}

• http://mi.mathnet.ru/eng/tvp731
• http://mi.mathnet.ru/eng/tvp/v12/i3/p506

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This publication is cited in the following articles:
1. M. A. Lifshits, Ya. Yu. Nikitin, V. V. Petrov, A. Yu. Zaitsev, A. A. Zinger, “Toward the history of the Saint Petersburg school of probability and statistics. I. Limit theorems for sums of independent random variables”, Vestn. St Petersb. Univ. Math., 51:2 (2018), 144–163