Bernoulli

  • Bernoulli
  • Volume 8, Number 5 (2002), 627-642.

Irregular sets and central limit theorems

Gonzalo Perera

Full-text: Open access

Abstract

In previous papers we have studied the asymptotic behaviour of$S_N(A;X)=(2N+1)^{-d/2}\Sigma_{n \in A_N}X_n$, where $X$ is a centred, stationary and weakly dependent random field, and $A_N=A \cap [ -N,N]^d, A \subset \mathbb {Z}^d$. This leads to the definition of asymptotically measurable sets, which enjoy the property that $S_N(A;X)$ 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 $X_n=\varphi (\xi_,n,Y_n), n \in \mathbb {Z}^d$, where $X_n$ is centred, $\varphi$ satisfies certain regularity conditions, and $\xi$ and $Y$ are independent random fields; for any $m \in \mathbb {N}$ and $(y1,\ldots,y_m)$, the central limit theorem holds for ($\varphi (\xi ,y_1),\ldots, \xi ,y_m)$), but $Y$ satisfies only the strong law of large numbers as it applies to $(Y_m,Y_{m-n})_{m \in \mathbb{Z}^d}$, for any $n \in \mathbb{Z}^d$. Under these conditions, it is shown that the central limit theorem holds for $X$.

Article information

Source
Bernoulli, Volume 8, Number 5 (2002), 627-642.

Dates
First available in Project Euclid: 4 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1078435221

Mathematical Reviews number (MathSciNet)
MR2003h:60038

Zentralblatt MATH identifier
1014.60016

Keywords
asymptotically measurable collections of sets central limit theorems level sets regression models weakly dependent random fields

Citation

Perera, Gonzalo. Irregular sets and central limit theorems. Bernoulli 8 (2002), no. 5, 627--642. https://projecteuclid.org/euclid.bj/1078435221


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