CLT, MDP and LDP for Range-Renewals of I.I.D.Samplings from an Infinite Discrete Distribution

Let $R_n$ be the number of distinct values of the $n$ simple samples from an infinite discrete distribution. In 1960 Bahadure proved ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{R_n}{\text{Enum}{R}_n}=1}$ 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.