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

TL;DR

This paper establishes central limit, moderate deviation, and large deviation principles for the number of distinct values in i.i.d. samples from an infinite discrete distribution, extending classical convergence results.

Contribution

It introduces CLT, MDP, and LDP results for range-renewals of i.i.d. samples, providing a deeper probabilistic understanding beyond existing convergence theorems.

Findings

CLT for the number of distinct values $R_n$

MDP results under mild conditions

LDP results for $R_n$

Abstract

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.