On the rates in the central limit theorem for weakly dependent random fields

SummaryLet {Xa;a∈Zd} (d≧2) be a random field satisfying some weak dependence condition. For a finite subset V of Zd, set $$S(V) = \sum\limits_{a \in V} {X_a } $$ . In this paper, under the conditions related to moment and dependence coefficients, we show that L∞- and L1-rates in the central limit theorem for S(V) are of order O(¦V¦−1/2(log¦V¦)d) (strong mixing case): O(¦V¦−1/2) (m-dependent case). Here ¦V¦ denotes the number of elements in V. The content of this paper is a negative answer to the conjecture of Prakasa Rao (Z. Wahrscheinlichkeitstheorie verw. Gebiete 58, 247–256 (1981)).

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