Large Deviation Probabilities for Samples from a Finite Population

Abstract

Let $X_n$ be the standardized mean of $s$ observations obtained by simple random sampling from the $n$ numbers $a_{n1},\cdots, a_{nn}$ and let $b_n$ be the maximum deviation of these numbers from their mean. If $b_n$ tends to zero then the distribution function of $X_n$ tends uniformly to the normal distribution function. However this approximation is not adequate at the tails of the distribution. Here we obtain limit theorems for $P(X_n > x)$ in the two cases when $x = o(b_n^{-1})$ and $x = O(b_n^{-1})$. These are related to similar results for sums of independent random variables.