Nonconventional large deviations theorems

We obtain large deviations theorems for both discrete time expressions of the form ∑_n=1^N F(X(q1(n)),..., X(qℓ(n))) and similar expressions of the form ∫₀^T F(X(q1(t)),..., X(qℓ(t))) dt in continuous time. Here X(n),n≥ 0 or X(t), t≥ 0 is a Markov process satisfying Doeblin's condition, F is a bounded continuous function 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. Applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations will be obtained, as well.