Berry–Esseen type estimates for nonconventional sums

Abstract

We obtain Berry-Esseen type estimates for “nonconventional” expressions of the form ξN=1N∑n=1N(F(X(q1(n)),…,X(qℓ(n)))−F̄) where X(n) is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties, F̄=∫Fd(μ×⋯×μ), μ is the distribution of X(0) and qi(n)=in for 1≤i≤k while for i>k they are positive functions taking integer values with some growth conditions which are satisfied, for instance, when they are polynomials of increasing degrees. Our setup is similar to Kifer and Varadhan (2014) where a nonconventional functional central limit theorem was obtained and the present paper provides estimates for the convergence speed. As a part of the study we provide answers for the crucial question on positivity of the limiting variance limN→∞Var(ξN) which was not studied in Kifer and Varadhan (2014). Extensions to the continuous time case will be discussed as well. As in Kifer and Varadhan (2014) our results are applicable to stationary processes generated by some classes of sufficiently well mixing Markov chains and dynamical systems.