Nonconventional polynomial CLT

Abstract

We obtain a functional central limit theorem (CLT) for sums of the form , where 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, is a certain centralizing constant and are arbitrary polynomials taking on positive integer values on positive integers, i.e. polynomials satisfying , where is the set of natural numbers. For polynomial ’s this CLT generalizes [The Annals of Probability, 2014, Vol. 42, No.2, 649–688] which allows only linear ’s to have the same polynomial degree. We also prove that exists and provide necessary and sufficient conditions for its positivity, which is equivalent to the statement that the weak limit of 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 , which is done in a separate section. As in [The Annals of Probability, 2014, Vol. 42, No.2, 649-688], our results hold true when , 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 , where is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.