Limit Theorems for Multivariate Lacunary Systems

TL;DR

This paper extends recent probabilistic limit theorems for lacunary systems from one-dimensional to multivariate cases, using a novel martingale approximation approach that bypasses Fourier analysis.

Contribution

It develops a new method to prove the Central Limit Theorem and Law of the Iterated Logarithm for multivariate lacunary systems, expanding the scope of previous one-dimensional results.

Findings

Established CLT for multivariate lacunary systems

Proved LIL in the multivariate setting

Introduced a martingale approximation technique

Abstract

Lacunary function systems of type $(f(M_nx))_{n\geq 1}$ for periodic functions $f$ and sequences of fast-growing matrices $(M_n)_{n\geq 1}$ exhibit many properties of independent random variables like satisfying the Central Limit Theorem or the Law of the Iterated Logarithm. It is well-known that this behaviour depends on number theoretic properties of $(M_n)_{n\geq 1}$ as well as analytic properties of $f$. Classical techniques are essentially based on Fourier analysis making it almost impossible to use a similar approach in the multivariate setting. Recently Aistleitner and Berkes introduced a new method proving the Central Limit Theorem in the one-dimensional case by approximating $\sum_{n}f(M_nx)$ by a sum of piecewise constant periodic functions which form a martingale differences sequence and using a Berry-Esseen type inequality. Later this approach was used to show the Law of the Iterated Logarithm by a consequence of Strassen's almost sure invariance principle. In this paper we develop this method to prove the Central Limit Theorem and the Law of the Iterated Logarithm in the multidimensional case.