Nonconventional limit theorems

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 ∑_n=1^N T^q₁⁽ⁿ⁾ f₁... T^{qe (n)} 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√N ∑_n=1^N (F(X₀(n),X₁(q₁(n)),X₂(q₂(n)), ..., X_e(q_e(n)))F̄) where X_i's are exponentially fast ψ-mixing bounded processes with some stationarity properties, F is a Lipschitz continuous function, F̄ = ∫ Fd(μ ₀ × μ ₁ × ... × μ _l), 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(x₀, x₁, . . ., x_ℓ) = x₀x₁x₂ . . . x_ℓ exponentially fast α-mixing already suffices. This result can be applied in the case when X_i(n) = Tⁿ 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.