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May  2020, 40(5): 2827-2873. doi: 10.3934/dcds.2020151

A functional CLT for nonconventional polynomial arrays

 Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

* Corresponding author: Yeor Hafouta

Received  July 2019 Revised  December 2019 Published  March 2020

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).
Citation: Yeor Hafouta. A functional CLT for nonconventional polynomial arrays. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2827-2873. doi: 10.3934/dcds.2020151
References:

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