Precise Limit Theorems for Lacunary Series

TL;DR

This paper establishes precise limit theorems, including the central limit theorem and local limit theorems, for lacunary series and H"older continuous functions, using martingale theory and mod-Gaussian convergence.

Contribution

It introduces an alternative martingale-based approach to prove limit theorems for lacunary series, extending results to infinite product spaces and providing bounds and deviations.

Findings

Proved central limit theorem for lacunary series with big gaps.

Established local limit theorems and Berry-Esseen bounds.

Identified scales where Gaussian approximation fails.

Abstract

Lacunary trigonometric and Walsh series satisfy limiting results that are typical for i.i.d. random variables such as the central limit theorem (Salem, Zygmund 1947), the law of the iterated logarithm (Weiss 1959) and several probability related limit theorems. For H\"older continuous, periodic functions this phenomenon does not hold in general. Kac (1946, 1949) showed the validity of the central limit theorem for the sequence $\left(f(2^k x)\right)_k$ and in the case of "big gaps''. In this paper, we present an alternative approach to prove the above theorem based on martingale theory, which allows us to generalize the theorem to infinite product spaces of arbitrary probability spaces, equipped with the shift operator. In addition, we show the local limit theorems for lacunary trigonometric and Walsh series, and for H\"older continuous, periodic functions in the case of "big gaps''. We also establish Berry-Esseen bounds and moderate deviations for lacunary Walsh series. Furthermore, we identify the scale at which the validity of the Gaussian approximation for the tails breaks. To derive these limiting results, the framework of mod-Gaussian convergence has been used.