A law of the iterated logarithm for general lacunary series
Volume 126 / 2012
Colloquium Mathematicum 126 (2012), 95-103
MSC: Primary 42A55; Secondary 60F15.
DOI: 10.4064/cm126-1-6
Abstract
We prove a law of the iterated logarithm for sums of the form $\sum_{k=1}^N a_k f(n_k x)$ where the $n_k$ satisfy a Hadamard gap condition. Here we assume that $f$ is a Dini continuous function on $\mathbb R^n$ which has the property that for every cube $Q$ of sidelength 1 with corners in the lattice $\mathbb Z^n$, $f$ vanishes on $\partial Q$ and has mean value zero on $Q.$
