Nonconventional limit theorems

Abstract

The polynomial ergodic theorem (PET) which appeared in Bergelson (Ergod. Th. Dynam. Sys. 7, 337–349, 1987) and attracted substantial attention in ergodic theory studies the limits of expressions having the form $${1/N\sum_{n=1}^NT^{q_1(n)}f_1 \cdots T^{q_\ell (n)} f_\ell}$$ where T is a weakly mixing measure preserving transformation, f i ’s are bounded measurable functions and q i ’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 $${1/\sqrt{N}\sum_{n=1}^N (F(X_0(n),X_1(q_1(n)),X_2(q_2(n)), \ldots, X_\ell(q_\ell(n)))-\overline F)}$$ where X i ’s are exponentially fast ψ-mixing bounded processes with some stationarity properties, F is a Lipschitz continuous function, $${\overline F=\int Fd(\mu_0\times\mu_1\times \cdots\times\mu_\ell)}$$ , μ j is the distribution of X j (0), and q i ’s are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q i ’s are polynomials of growing degrees. When F(x 0, x 1, . . . , x l ) = x 0 x 1 x 2 . . . x l exponentially fast α-mixing already suffices. This result can be applied in the case when X i (n) = T n f 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 (ξ n ) where ξ n is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.