  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.
