How Many Digits are Needed?

Abstract

Let \(X_1,X_2,...\) be the digits in the base-q expansion of a random variable X defined on [0, 1) where \(q\ge 2\) is an integer. For \(n=1,2,...\), we study the probability distribution \(P_n\) of the (scaled) remainder \(T^n(X)=\sum _{k=n+1}^\infty X_k q^{n-k}\): If X has an absolutely continuous CDF then \(P_n\) converges in the total variation metric to the Lebesgue measure \(\mu \) on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that \(T^N(X)\) follows \(\mu \) and is independent of \((X_1,...,X_N)\), and second exponentially fast convergence of \(P_n\) and its PDF \(f_n\). We discuss how many digits are needed and show examples of our results.