О нормальной аппроксимации для случайных полей с сильным перемешиванием

Let ξn be a strongly mixing sequence of real random variables such that Eξn = 0. Write Sn = ξ1 + · · ·+ ξn and consider the normalized sums Zn = Sn/Bn, where B n = ES 2 n. Assume that a thrice differentiable function h : R → R satisfies supx∈R |h (x)| < ∞. We obtain optimal (in a sense) bounds for ∆n = |Eh(Zn)−Eh(N)|, where N is a standard normal random variable. Namely, we show that ∆n = O(n ), provided that the random variables ξn are bounded by a constant, B 2 n ≥ c0n, where c0 is a positive constant, and that the strong mixing coefficients α(r) satisfy P ∞ r=1 rα(r) < ∞. The results extend to the case of random fields {ξa, a ∈ Z }. To prove the results we apply a new method.