Functional limit theorems for digital expansions

Abstract

The main purpose of this paper is to discuss the asymptotic behaviour of the difference sq,k(P(n)) - k(q-1)/2 where sq,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 q1-ary and q2-ary digital expansions where q1 and q2 are coprime.