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Irregular Sets and Central Limit Theorems

Gonzalo Perera
Bernoulli
Vol. 8, No. 5 (Oct., 2002), pp. 627-642
Stable URL: http://www.jstor.org/stable/3318949
Page Count: 16
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Irregular Sets and Central Limit Theorems
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Abstract

In previous papers we have studied the asymptotic behaviour of SN(A;X)=(2N+1)-d/2∑n∈ AN Xn, where X is a centred, stationary and weakly dependent random field, and AN=A∩ [-N,N]d, $A\subset {\Bbb Z}^{d}$. This leads to the definition of asymptotically measurable sets, which enjoy the property that SN(A;X) has a (Gaussian) weak limit for any X belonging to a certain class. We present here an application of this technique. Consider a regression model Xn=φ (ξ n,Yn), n∈ Zd, where Xn is centred, φ satisfies certain regularity conditions, and ξ and Y are independent random fields; for any m∈ N, and (y1,... ,ym) the central limit theorem holds for (φ (ξ ,y1),...,φ (ξ ,ym)), but Y satisfies only the strong law of large numbers as it applies to (Ym,Ym-n)m∈ Zd , for any n∈ Zd. Under these conditions, it is shown that the central limit theorem holds for X.

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