Nonconventional large deviations theorems

We obtain large deviations theorems for both discrete time expressions of the form and similar expressions of the form in continuous time. Here or is a Markov process satisfying Doeblin’s condition, is a bounded continuous function and for while for they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when ’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.