On the central limit theorem for f (n k x)

Abstract

By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) n≥1 with 2π -periodic f ∈ L 2 this phenomenon is generally not valid and the asymptotic behavior of ( f (n k x)) k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n k . By the classical theory, the central limit theorem holds for f (n k x) if n k = 2 k , or if n k+1 /n k → α with a transcendental α, but it fails e.g., for n k = 2 k − 1. The purpose of our paper is to give a necessary and sufficient condition for f (n k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened "infinitesimally".

Irregular discrepancy behavior of lacunary series

“…In our recent paper [1] we proved that if (n k ) k≥1 satisfies a number theoretic condition slightly stronger than what is required for the validity of the central limit theorem for N k=1 f (n k x), see [2], the limsup in (1) is equal to 1/2. This covers, for example, the case when n k+1 /n k → θ, where θ r is irrational for r = 1, 2, .…”

Convergence of $∑c_{k} f(k x)$ and the Lip $𝛼$ class

“…If the condition that f ∈ Lip α for some α > 1/2 is dropped, the convergence of the sum (1) will in general no longer be sufficient to guarantee the a.e. convergence of (2). One possibility to meet this fact is to consider series of the form…”

On the law of the iterated logarithm for the discrepancy of lacunary sequences

“…In our recent paper [1] we showed that f (n k x) satisfies the CLT for all "nice" periodic functions f provided for any d ≥ 1,…”