How Many Digits are Needed?

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