A functional CLT for nonconventional polynomial arrays

In this paper we will prove a functional central limit theorem (CLT) for random functions of the form

$ {\mathcal S}_N(t) = N^{-\frac12}\sum\limits_{n = 1}^{[Nt]} F(\xi_{q_1(n, N)}, \xi_{q_2(n, N)}, ..., \xi_{q_\ell(n, N)}) $

where the $ q_i $'s are certain type of bivariate polynomials, $ F = F(x_1, ..., x_\ell) $ is a locally Hölder continuous function and the sequence of random variables $ \{\xi_n\} $ satisfies some mixing and moment conditions. This paper continues the line of research started in [15] and [17], and it is a generalization of the results in [9] and Chapter 3 of [11]. We will also prove a strong law of large numbers (SLLN) for the averages $ N^{-\frac12} {\mathcal S}_N(1) $ which extends the results from the beginning of Chapter 3 of [11] to general bivariate polynomial functions $ q_i $. Our results hold true for sequences $ \{\xi_n\} $ generated by a wide class of Markov chains and dynamical systems. As an application we obtain functional CLT's for expressions of the form $ N^{-\frac12}M([Nt]) $, where $ M(N) $ counts the number of multiple recurrence of the sequence $ \{\xi_n\} $ to certain sets $ A_1, ..., A_\ell $ which occur at the times $ q_1(n, N), ..., q_\ell(n, N) $, as well as SLLN's for these $ M(N) $'s. One of the simplest examples is when $ \xi_n $ is $ n $-the digit of a random $ m $-base or continued fraction expansion, and each $ A_i $ is singleton (i.e. it represent one possible value of a digit).