How Many Digits are Needed?

Abstract

AbstractLet $$X_1,X_2,...$$ 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 ≥ 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 ) = ∑ k = n + 1 ∞ 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 $$\mu $$ μ 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 $$\mu $$ μ 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.