Functional limit theorems for digital expansions

Authors: Drmota, M.; Fuchs, M.; Manstavičius, E.

Source: Acta Mathematica Hungarica, Volume 98, Number 3, February 2002 , pp. 175-201(27)

Publisher: Springer

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Abstract:

The main purpose of this paper is to discuss the asymptotic behaviour of the difference s _q,k(P(n)) - k(q-1)/2 where s _q,k (n) denotes the sum of the first k digits in the q-ary digital expansion of n and P(x) is an integer polynomial. We prove that this difference can be approximated by a Brownian motion and obtain under special assumptions on P, a Strassen type version of the law of the iterated logarithm. Furthermore, we extend these results to the joint distribution of q ₁-ary and q ₂-ary digital expansions where q ₁ and q ₂ are coprime.

Keywords: sum-of-digits function; functional limit theorem; Strassen's law of the iterated logarithm; q-ary digital expansion

Document Type: Research article

DOI: http://dx.doi.org/10.1023/A:1022869708089

Publication date: 2002-02-01