Radix expansions and the uniform distribution

Let the random variable X be uniformly distributed on [0,1], [alpha] be a positive number, [alpha][not equal to]1, and b be a positive integer, b>1. We derive the joint distribution of Y1,Y2,...,Yk, the first k significant digits in the radix expansion in base b of Y=X1/[alpha]. We show that, as k-->[infinity], Yk converges in distribution to the uniform distribution on the set {0,1,...,b-1}. We also prove that if Y is a random variable taking values in [0,1] whose cumulative distribution function is continuous and convex (respectively, concave) then the significant digits Y1,Y2,... are stochastically increasing (respectively, decreasing). In particular, if Y=X1/[alpha] where X is uniformly distributed on [0,1] then the significant digits Y1,Y2,... are stochastically increasing (respectively, decreasing) if [alpha] 1).