The Law of the Iterated Logarithm

Abstract

The article begins first with the history and the development of the law of the iterated logarithm, abbreviated LIL. We then discuss the LIL in the context of independent random variables, dyadic martingales, lacunary trigonometric series, and harmonic functions. Finally, we derive a LIL for a sequence of dyadic martingales.

An Upper Bound in a Law of the Iterated Logarithm for Rademacher Function

“…The former case is commonly called regular LIL whereas the latter law is called the tail LIL. For more about the law of the iterated logarithm, the reader is referred to [1], [3], [5], and [6]. Salem and Zygmund achieved similar LIL for series of lacunary which is taken as the opening result in such an area of mathematics.…”