EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS

Götze F, Zaitsev AY (2014)
The Annals of Probability 42(1): 354-397.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
Let X, X-1, X-2,... be i.i.d. R-d-valued real random vectors. Assume that EX = 0, cov X = C, E vertical bar vertical bar X vertical bar vertical bar(2) = sigma(2) and that X is not concentrated in a proper subspace of R-d Let G be a mean zero Gaussian random vector with the same covariance operator as that of X. We study the distributions of nondegenerate quadratic forms Q[S-N] of the normalized sums S-N = N-(1/2) (X-1 +...+ X-N) and show that, without any additional conditions, Delta(N) =(def) sup(x)[P{Q[S-N] <= x} - P {Q[G] <= x} vertical bar = O(N-1) provided that d >= 5 and the fourth moment of X exists. Furthermore, we provide explicit bounds of order O(N-1) for AN for the rate of approximation by short asymptotic expansions and for the concentration functions of the random variables Q[S-N + a], a epsilon R-d. The order of the bound is optimal. It extends previous results of Bentkus and Glaze [Probab. Theory Related Fields 109 (1997a) 367-416] (for d >= 9) to the case d >= 5, which is the smallest possible dimension for such a bound. Moreover, we show that, in the finite dimensional case and for isometric Q, the implied constant in O(N-1) has the form c(d) sigma(d) (detC)-E-1/2 vertical bar vertical bar C-1/2 X vertical bar vertical bar(4) with some cd depending on d only. This answers a long standing question about optimal rates in the central limit theorem for quadratic forms starting with a seminal paper by Esseen [Acta Math. 77 (1945) 1-125].
Erscheinungsjahr
Zeitschriftentitel
The Annals of Probability
Band
42
Ausgabe
1
Seite(n)
354-397
ISSN
PUB-ID

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Götze F, Zaitsev AY. EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS. The Annals of Probability. 2014;42(1):354-397.
Götze, F., & Zaitsev, A. Y. (2014). EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS. The Annals of Probability, 42(1), 354-397. doi:10.1214/13-AOP839
Götze, F., and Zaitsev, A. Y. (2014). EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS. The Annals of Probability 42, 354-397.
Götze, F., & Zaitsev, A.Y., 2014. EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS. The Annals of Probability, 42(1), p 354-397.
F. Götze and A.Y. Zaitsev, “EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS”, The Annals of Probability, vol. 42, 2014, pp. 354-397.
Götze, F., Zaitsev, A.Y.: EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS. The Annals of Probability. 42, 354-397 (2014).
Götze, Friedrich, and Zaitsev, Andrei Yu. “EXPLICIT RATES OF APPROXIMATION IN THE CLT FOR QUADRATIC FORMS”. The Annals of Probability 42.1 (2014): 354-397.