  Let $R_n$ be the number of distinct values of the $n$ simple samples from an
infinite discrete distribution. In 1960 Bahadure proved $\displaystyle
\lim_{n\to \infty} \frac{R_n}{\Enum R_n}=1$ in probability; Chen et al. proved
the limit in the sense of almost sure convergence, along with other results. In
this note we present results of CLT, MDP and LDP for $R_n$ under mild
conditions.
