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Teoriya Veroyatnostei i ee Primeneniya, 1967, Volume 12, Issue 4, Pages 655–665 (Mi tvp752)  

This article is cited in 19 scientific papers (total in 19 papers)

The central limit theorem for sums of functions of independent random variables and for sums of the form $\sum f(2^kt)$

I. A. Ibragimov

Leningrad
Abstract: Let $\varepsilon_1,\varepsilon_2,\dots$ be a sequence of independent random variables and let a random variable $f=f(\varepsilon_1,\varepsilon_2,\dots)$. Consider a sequence of random variables $\{f_j\}$ where $f_j=f(\varepsilon_j,\varepsilon_{j+1},\dots)$. The main result of this paper is
Theorem 2. {\it If
$1)\ \mathbf E|f|^{2+\delta}=\rho_\delta<\infty$ for some $\delta$, $0<\delta\le1$;
$2)\ \mathbf E^{\frac1{2+\delta}}|f-\mathbf E\{f\mid\varepsilon_1,\dots,\varepsilon_n\}|^{2+\delta}\le A2^{-n\alpha}$ where $A$, $\alpha$ are positive constants;
$3)\ \sigma^2=\mathbf Ef_1^2+2\sum_2^\infty\mathbf E\{f_1f_j\}\ne0$ then
$$ \biggl|\mathbf P\biggl\{\frac1{\sigma\sqrt n}\sum_1^nf_j<z\biggr\}-\Phi(z)\biggr|\le C\biggl(\frac{\ln n}{n}\biggr)^{\delta/2}, $$
where $\Phi(z)=\frac1{\sqrt{2\pi}}\int_{-\infty}^ze^{-\frac{x^2}{2}}\,dx$ and $C$ depends on $A$, $\alpha$, $\sigma$, $\rho_\delta$ only}.
This theorem is applied to find distributions of sums $\sum_1^\infty f(2^kt)$.
Received: 15.12.1966
English version:
Theory of Probability and its Applications, 1967, Volume 12, Issue 4, Pages 596–607
DOI: https://doi.org/10.1137/1112075
Bibliographic databases:
Language: Russian
Citation: I. A. Ibragimov, “The central limit theorem for sums of functions of independent random variables and for sums of the form $\sum f(2^kt)$”, Teor. Veroyatnost. i Primenen., 12:4 (1967), 655–665; Theory Probab. Appl., 12:4 (1967), 596–607
Citation in format AMSBIB
\Bibitem{Ibr67}
\by I.~A.~Ibragimov
\paper The central limit theorem for sums of functions of independent random variables and for sums of the form $\sum f(2^kt)$
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 4
\pages 655--665
\mathnet{http://mi.mathnet.ru/tvp752}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=226711}
\zmath{https://zbmath.org/?q=an:0217.49803}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 4
\pages 596--607
\crossref{https://doi.org/10.1137/1112075}
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  • https://www.mathnet.ru/eng/tvp/v12/i4/p655
  • This publication is cited in the following 19 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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