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Mat. Sb., 1989, Volume 180, Number 12, Pages 1587–1613 (Mi msb1677)  

This article is cited in 6 scientific papers (total in 6 papers)

A precise estimate of the rate of convergence in the Central Limit Theorem in Hilbert space

B. A. Zalesskii, V. V. Sazonov, V. V. Ulyanov


Abstract: Let
$$ S_n=n^{-1/2}\sigma^{-1}\sum_1^n(X_i-\mathbf EX_i),\quad\sigma^2=\mathbf E|X_1-\mathbf EX_1|^2, $$
be the normed sum of independent identically distributed random variables $X_i$ with values in a separable Hilbert space $H$. Denote by $V$ the covariance operator of $X$, and let $Y$ be an $H$-valued $(0,\sigma^{-2}V)$ Gaussian random variable. The authors prove that there exist an absolute constant such that for any $a\in H$ and $r\geqslant0$
$$ |\mathbf P(|S_n-a|<r)-\mathbf P(|Y-a|<r)|\leqslant c(\prod_1^6\sigma_i^{-1})\sigma^3\mathbf E|X_1-\mathbf EX_1|^3(1+|a|^3)n^{-1/2}, $$
where $\sigma_1^2\geqslant\sigma_2^2\geqslant\dotsb$ are the eigenvalues of $V$. Up to the value of $c$, this estimate is unimprovable in general.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Sbornik, 1991, 68:2, 453–482

Bibliographic databases:

UDC: 519.2
MSC: 60B12, 60F05
Received: 16.01.1989

Citation: B. A. Zalesskii, V. V. Sazonov, V. V. Ulyanov, “A precise estimate of the rate of convergence in the Central Limit Theorem in Hilbert space”, Mat. Sb., 180:12 (1989), 1587–1613; Math. USSR-Sb., 68:2 (1991), 453–482

Citation in format AMSBIB
\Bibitem{ZalSazUly89}
\by B.~A.~Zalesskii, V.~V.~Sazonov, V.~V.~Ulyanov
\paper A~precise estimate of the rate of convergence in the Central Limit Theorem in Hilbert~space
\jour Mat. Sb.
\yr 1989
\vol 180
\issue 12
\pages 1587--1613
\mathnet{http://mi.mathnet.ru/msb1677}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1038219}
\zmath{https://zbmath.org/?q=an:0694.60004|0709.60006}
\transl
\jour Math. USSR-Sb.
\yr 1991
\vol 68
\issue 2
\pages 453--482
\crossref{https://doi.org/10.1070/SM1991v068n02ABEH002110}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1991FE73700008}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. V. Sazonov, V. V. Ulyanov, “Asymptotic expansions of the probability that the sum of independent random variables hits a ball in a Hilbert space”, Russian Math. Surveys, 50:5 (1995), 1045–1063  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Bentkus V., Gotze F., “Optimal Rates of Convergence in the Clt for Quadratic Forms”, Ann. Probab., 24:1 (1996), 466–490  crossref  mathscinet  zmath  isi
    3. Yu. V. Borovskikh, L. Madan Puri, V. V. Sazonov, “Normal Approximation of U-Statistics in Hilbert Space”, Theory Probab Appl, 41:3 (1997), 405  mathnet  crossref  mathscinet  isi
    4. A. N. Nazarova, “Logarifmicheskaya skorost skhodimosti v TsPT dlya sluchainykh lineinykh protsessov i polei v gilbertovom prostranstve”, Fundament. i prikl. matem., 8:4 (2002), 1091–1098  mathnet  mathscinet  zmath  elib
    5. S. A. Bogatyrev, “A Nonuniform Estimate for the Error in Short Asymptotic Expansions in Hilbert Space”, Theory Probab Appl, 47:4 (2003), 689  mathnet  crossref  mathscinet  zmath  isi  elib
    6. S. V. Nagaev, V. I. Chebotarev, “On the Accuracy of Gaussian Approximation in Hilbert Space”, Siberian Adv. Math., 15:1 (2005), 11–73  mathnet  mathscinet  zmath  elib
  • Математический сборник - 1989–1990 Sbornik: Mathematics (from 1967)
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