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Statistical independence in mathematics – the key to a Gaussian law. (English) Zbl 1465.60023

Summary: In 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 P. Erdős and M. Kac [Am. J. Math. 62, 738–742 (1940; JFM 66.0172.02; Zbl 0024.10203)] on the number of different prime factors of a number \(n\in{\mathbb{N}} \). We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of R. Salem and A. Zygmund [Proc. Natl. Acad. Sci. USA 33, 333–338 (1947; Zbl 0029.11902)].

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
11K65 Arithmetic functions in probabilistic number theory
42A55 Lacunary series of trigonometric and other functions; Riesz products
42A61 Probabilistic methods for one variable harmonic analysis
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