On the distribution of the van der Corput sequence in arbitrary base

Abstract

A central limit theorem with explicit error bound, and a large deviation result are proved for a sequence of weakly dependent random variables of a special form. As a corollary, under certain conditions on the function $f:[0,1]\to \mathbb{R}$ a central limit theorem and a large deviation result are obtained for the sum ${\sum }_{n=0}^{N-1}f(x_n)$, where $x_n$ is the base $b$ van der Corput sequence for an arbitrary integer $b\ge 2$. Similar results are also proved for the $L^p$ discrepancy of the same sequence for $1\le p<\infty$. The main methods used in the proofs are the Berry-Esseen theorem and Fourier analysis.

arXiv comment26 pages, minor changes