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 Zap. Nauchn. Sem. POMI, 2010, Volume 384, Pages 105–153 (Mi znsl3887)

Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms

F. Götzea, A. Yu. Zaitsevb

a Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany
b St. Petersburg Branch Steklov Mathematical Institute, St. Petersburg, Russia

Abstract: Let $X,X_1,X_2,…$ be i.i.d. $\mathbb R^d$-valued real random vectors. Assume that $\mathbf EX=0$ and that $X$ has a non-degenerate distribution. Let $G$ be a mean zero Gaussian random vector with the same covariance operator as that of $X$. We investigate the distributions of non-degenerate quadratic forms $\mathbb Q[S_N]$ of the normalized sums $S_N=N^{-1/2}(X_1+…+X_N)$ and show that, without any additional conditions, for any $a\in\mathbb R^d$,
$$\Delta_N^{(a)}\stackrel{\mathrm{def}}=\sup_x|\mathbf P\{\mathbb Q[S_N-a]\le x\}-\mathbf P\{\mathbb Q[G-a]\le x\}-E_a(x)|=\mathcal O(N^{-1}),$$
provided that $d\ge5$ and $\mathbf E\|X\|^4<\infty$. Here $E_a(x)$ is the Edgeworth type correction of order $\mathcal O(N^{-1/2})$. Furthermore, we provide explicit bounds of order $\mathcal O(N^{-1})$ for $\Delta_N^{(a)}$ and for the concentration function of the random variable $\mathbb Q[S_N+a]$, $a\in\mathbb R^d$. Our results extend the corresponding results of Bentkus and Götze (1997) ($d\ge9$) to the case $d\ge5$. Bibl. 35 titles.

Key words and phrases: Central Limit Theorem, quadratic forms, concentration inequalities, convergence rates.

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English version:
Journal of Mathematical Sciences (New York), 2011, 176:2, 162–189

UDC: 519

Citation: F. Götze, A. Yu. Zaitsev, “Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms”, Probability and statistics. Part 16, Zap. Nauchn. Sem. POMI, 384, POMI, St. Petersburg, 2010, 105–153; J. Math. Sci. (N. Y.), 176:2 (2011), 162–189

Citation in format AMSBIB
\Bibitem{GotZai10} \by F.~G\"otze, A.~Yu.~Zaitsev \paper Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms \inbook Probability and statistics. Part~16 \serial Zap. Nauchn. Sem. POMI \yr 2010 \vol 384 \pages 105--153 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl3887} \transl \jour J. Math. Sci. (N. Y.) \yr 2011 \vol 176 \issue 2 \pages 162--189 \crossref{https://doi.org/10.1007/s10958-011-0408-5} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79959559903} 

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This publication is cited in the following articles:
1. Goetze F., Zaitsev A.Yu., “Explicit Rates of Approximation in the Clt for Quadratic Forms”, Ann. Probab., 42:1 (2014), 354–397
2. I. S. Borisov, N. V. Volod'ko, “Asymptotic expansions for the distributions of canonical $V$-statistics of third order”, Theory Probab. Appl., 60:1 (2016), 1–18
3. Lifshits M.A. Nikitin Ya.Yu. Petrov V.V. Zaitsev A.Yu. Zinger A.A., “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
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