Stein's method for nonconventional sums

We obtain almost optimal convergence rate in the central limit theorem for (appropriately normalized) "nonconventional" sums of the form S-N = Sigma(N)(n=1) (F(xi(n),xi(2n), ... ,xi l(n)) - (F) over bar). Here {xi(n) : n >= 0} is a sufficiently fast mixing vector process with some stationarity conditions, F is bounded Holder continuous function and (F) over bar is a certain centralizing constant. Extensions to more general functions F will be discusses, as well. Our approach here is based on the so called Stein's method, and the rates obtained in this paper significantly improve the rates in [7]. Our results hold true, for instance, when xi(n) = (T-n f(i))(i = 1)(p) where T is a topologically mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when {xi(n) : n >= 0} forms a stationary and exponentially fast phi-mixing sequence, which, for instance, holds true when xi(n) = (f(i)((Upsilon)(n)))(i=1)(p) where Upsilon(n) is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.