Let $X$ have a uniform distribution on the interval $[0,1]$ and le $\mathrm{t} N_{m, k}$ be the digit in the $m$ th place to the right of the decim al point in $X^k$.
(a) Find $\lim _{m \rightarrow \infty} P\left(N_{m, m}=i\right)$ for $i=0,1,2, \cdots, 9$.
(b) Let $k(m)$ be a function of $m$, taking values greater than 1 .
Find a necessary and sufficient condition on $k(m)$ such that $\lim _{m \rightarrow \infty} P\left(N_{m, k(m)}=i\right)=\frac{1}{10}$ for $i=0,1, \cdots, 9$.