Statistical independence in mathematics–the key to a Gaussian law

Abstract

AbstractIn this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.