The Reverse of The Law of Large Numbers

TL;DR

This paper explores how increasing the number of outcomes in a fixed-size sample can cause the sample mean to diverge from the population mean, effectively reversing the traditional law of large numbers.

Contribution

It introduces conditions under which the variance of the sample mean increases with the number of outcomes, highlighting a reverse effect to the law of large numbers.

Findings

Variance of the sample mean can increase with more outcomes.

Sample mean divergence occurs under specific conditions.

Results are relevant for finite discrete random variable sampling.

Abstract

The Law of Large Numbers tells us that as the sample size (N) is increased, the sample mean converges on the population mean, provided that the latter exists. In this paper, we investigate the opposite effect: keeping the sample size fixed while increasing the number of outcomes (M) available to a discrete random variable. We establish sufficient conditions for the variance of the sample mean to increase monotonically with the number of outcomes, such that the sample mean ``diverges'' from the population mean, acting like an ``reverse'' to the law of large numbers. These results, we believe, are relevant to many situations which require sampling of statistics of certain finite discrete random variables.