Radix expansions and the uniform distribution

Let the random variable X be uniformly distributed on [0,1], α be a positive number, α≠1, and b be a positive integer, b>We derive the joint distribution of Y₁,Y₂,...,Y_k, the first k significant digits in the radix expansion in base b of Y=X^1/α. We show that, as k→∞,Y_k 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=X^1/α where X is uniformly distributed on [0,1] then the significant digits Y₁,Y₂,... are stochastically increasing (respectively, decreasing) if α<1 (respectively, α>1).