Nonconventional Limit Theorems

The polynomial ergodic theorem (PET) which appeared in [1] and attracted substantial attention in ergodic theory studies the limits of expressions having the form 1/N ∑n=1 T q1(n) f1 · · ·T ql(n) fl where T is a weakly mixing measure preserving transformation, fi’s are bounded measurable functions and qi’s are polynomials taking on integer values on the integers. Motivated partially by this result we obtain a central limit theorem for even more general expressions of the form 1/ √ N ∑n=1 ( F ( X0(n),X1(q1(n)),X2(q2(n)), · · · ,Xl(ql(n)) ) −F ) where Xi’s are exponentially fast ψ-mixing bounded processes with some stationarity properties, F is a Lipschitz continuous function, F = ∫ Fd(μ0 × μ1 × ·· · × μl), μ j is the distribution of X j(0), and qi’s are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when qi’s are polynomials of growing degrees. When F(x0,x1, ...,xl) = x0x1x2 · · ·xl exponentially fast α-mixing already suffices. This result can be applied in the case when Xi(n) = T n fi 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(ξn) where ξn is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.