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Geometry of $$\mathbb{Z}^d $$ and the Central Limit Theorem for Weakly Dependent Random Fields

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  • Gonzalo Perera

Abstract

We study the asymptotic distribution of $$S_N (A,X) = \sqrt {(2N + 1)} ^{ - d} (\sum {_{n \in A_N } } X_n )$$ where A is a subset of $$\mathbb{Z}^d $$ , A N = A∩[−N, N] d , v(A) = lim N card(A N) (2N+1) −d ∈(0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, S N(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if $$F_N (n;A) = {\text{card\{ }}A_N^c \cap {\text{(}}n + A_N {\text{)\} (}}2N + 1{\text{)}}^{ - d} $$ has a limit F(n; A) as N → ∞ for each $$n \in \mathbb{Z}^d $$ . We also study the class of sets A that satisfy this condition.

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  • Gonzalo Perera, 1997. "Geometry of $$\mathbb{Z}^d $$ and the Central Limit Theorem for Weakly Dependent Random Fields," Journal of Theoretical Probability, Springer, vol. 10(3), pages 581-603, July.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:3:d:10.1023_a:1022693309359
    DOI: 10.1023/A:1022693309359
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    References listed on IDEAS

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    1. Roussas, G. G., 1994. "Asymptotic Normality of Random Fields of Positively or Negatively Associated Processes," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 152-173, July.
    2. Bradley, Richard C., 1981. "Central limit theorems under weak dependence," Journal of Multivariate Analysis, Elsevier, vol. 11(1), pages 1-16, March.
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