One-Sided Version of Law of the Iterated Logarithm for Summations of Signum Functions

Abstract

The law of the iterated logarithm (LIL), which describes the rate of convergence for a convergent lacunary series, was established by R. Salem and A. Zygmund. This rate is determined based on the variance-like term of the remainder after $n$ terms of the series. In this article, we investigate a comparable one-sided LIL for sums of signum functions, which also relies on the remainder after $n$ terms.