Nonconventional polynomial CLT

We obtain a functional central limit theorem (CLT) for sums of the form (Formula presented.), where X(n), n ≥0 is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties, (Formula presented.) is a certain centralizing constant and Q_i, i ≥ 1 are arbitrary polynomials taking on positive integer values on positive integers, i.e. polynomials satisfying Q_i(N) ⊂ N, where N is the set of natural numbers. For polynomial Q_j ’s this CLT generalizes which allows only linear Q_j ’s to have the same polynomial degree. We also prove that D₂ = lim_N→∞Eξ²_N(1)exists and provide necessary and sufficient conditions for its positivity, which is equivalent to the statement that the weak limit of ξ_N is not zero almost surely. Finally, we study independence properties of the increments of the limiting process. Our proofs require studying asymptotic densities of special subsets of N, which is done in a separate section. As in [9], our results hold true when X_i(n) = Tⁿf_i, where T is a 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 X_i(n) = f_i(ϒ_n), where ϒn is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.