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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.
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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
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\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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http://mi.mathnet.ru/eng/msb1677 http://mi.mathnet.ru/eng/msb/v180/i12/p1587
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Bentkus V., Gotze F., “Optimal Rates of Convergence in the Clt for Quadratic Forms”, Ann. Probab., 24:1 (1996), 466–490
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