Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms |
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Authors: | F Götze A Yu Zaitsev |
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Institution: | (1) DMI, Univ. di Trieste, Via Valerio, 12/1, 34127 Trieste, Italy;(2) Applied Genomics Institute, Via Linussio, 51, 33100 Udine, Italy;(3) DIMI, Univ. di Udine, Via delle Scienze, 206, 33100 Udine, Italy |
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Abstract: | Let X, X
1, X
2,… be i.i.d.
\mathbbRd {\mathbb{R}^d} -valued real random vectors. Assume that E
X = 0 and that X has a nondegenerate distribution. 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
\mathbbQ SN ] \mathbb{Q}\left {{S_N}} \right] of the normalized sums S
N
= N
−1/2 (X
1 + ⋯ + X
N
) and show that, without any additional conditions,
DN(a) = supx | \textP{ \mathbbQ SN - a ] \leqslant x } - \textP{ \mathbbQ G - a ] \leqslant x } - Ea(x) | = O( N - 1 ) \Delta_N^{(a)} = \mathop {{\sup }}\limits_x \left| {{\text{P}}\left\{ {\mathbb{Q}\left {{S_N} - a} \right] \leqslant x} \right\} - {\text{P}}\left\{ {\mathbb{Q}\left {G - a} \right] \leqslant x} \right\} - {E_a}(x)} \right| = \mathcal{O}\left( {{N^{ - 1}}} \right) |
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Keywords: | |
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