Let X1,X2,... be the digits in the base-q expansion of a random
variable X defined on [0,1) where q≥2 is an integer. For n=1,2,...,
we study the probability distribution Pn of the (scaled) remainder
∑k=n+1∞Xqqn−k: If X has an absolutely continuous CDF then
Pn converges in the total variation metric to Lebesgue measure on the unit
interval; under certain smoothness conditions we establish exponentially fast
convergence of Pn and its PDF fn; and we give examples of these results.
The results are extended to the case of a multivariate random variable defined
on a unit cube.Comment: 15 pages, 2 figure