Necessary and Sufficient Condition for Asymptotic Standard Normality of the Two Sample Pivot

TL;DR

This paper establishes that Cesaro convergence of cross-sample correlations to zero is both necessary and sufficient for the asymptotic standard Normality of the two-sample pivot, regardless of sample size ratios or dependence structures.

Contribution

It provides a necessary and sufficient condition for the asymptotic normality of the two-sample pivot without restrictions on sample size ratios or dependence.

Findings

Cesaro convergence of cross-sample correlations is necessary and sufficient.

Iterated limits of the pivot are standard Normal.

Joint distribution convergence implies bivariate standard Normal.

Abstract

The asymptotic solution to the problem of comparing the means of two heteroscedastic populations, based on two random samples from the populations, hinges on the pivot underpinning the construction of the confidence interval and the test statistic being asymptotically standard Normal. The pivot is known to converge to the standard Normal distribution if the two samples are independent and the ratio of the sample sizes converges to a finite positive number. We show, without any restriction on the asymptotic behavior of the ratio of the sample sizes, that Cesaro convergence of the sequence of cross sample correlation coefficients to 0 is necessary and sufficient for the aforesaid pivotal convergence. We also obtain, without any assumption on the cross sample dependence structure, that both iterated limits of the pivot are standard Normal and if the joint distribution of the standardized sample means converges to a spherically symmetric distribution, then that distribution must be bivariate standard Normal.