SOME ESTIMATES OF THE NORMAL APPROXIMATION FOR φ-MIXING RANDOM VARIABLES

Let xi(n) be phi-mixing sequence of real random variables such that E xi(n) = 0, and let Y be a standard normal random variable. Write S-n = xi(1) + ... + xi(n) and consider the normalized sums Z(n) = S-n/ B-n, where B-n(2) = ESn2. Assume that a thrice differentiable function h : R -> R satisfies sup(x is an element of R) vertical bar h'''(x)vertical bar < infinity. We obtain upper bounds for Delta(n) = vertical bar Eh(Z(n)) - Eh(Y)vertical bar in terms of Lyapunov fractions with explicit constants (see Theorem 1). In a particular case, the obtained upper bound of Delta(n) is of order O(n(-1/2)). We note that the phi-mixing coefficients phi(r) are defined between the " past" and " future." To prove the results, we apply the Bentkus approach.