Nonconventional limit theorems in discrete and continuous time via martingales

We obtain functional central limit theorems for both discrete time expressions of the form 1/.N.[Nt]n=1(F(X(q1(n)),.,X(q.(n)))..F) and similar expressions in the continuous time where the sum is replaced by an integral. Here 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, .F=.Fd(.....), . is the distribution of X(0) and qi(n)=in for i.k.. while for i>k they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when qi.s are polynomials of increasing degrees. These results decisively generalize [Probab. Theory Related Fields 148 (2010) 71.106], whose method was only applicable to the case k=2 under substantially more restrictive moment and mixing conditions and which could not be extended to convergence of processes and to the corresponding continuous time case. As in [Probab. Theory Related Fields 148 (2010) 71.106], our results hold true when Xi(n)=Tnfi, 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 Xi(n)=fi(.n), where .n is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure. Moreover, our relaxed mixing conditions yield applications to other types of dynamical systems and Markov processes, for instance, where a spectral gap can be established. The continuous time version holds true when, for instance, Xi(t)=fi(.t), where .t is a nondegenerate continuous time Markov chain with a finite state space or a nondegenerate diffusion on a compact manifold. A partial motivation for such limit theorems is due to a series of papers dealing with nonconventional ergodic averages.