  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, which is known to
happen if the two samples are independent and the ratio of the sample sizes
converges to a finite positive number. This restriction on the asymptotic
behavior of the ratio of the sample sizes carries the risk of rendering the
asymptotic justification of the finite sample approximation invalid. It turns
out that neither the restriction on the asymptotic behavior of the ratio of the
sample sizes nor the assumption of cross sample independence is necessary for
the pivotal convergence in question to take place. If the joint distribution of
the standardized sample means converges to a spherically symmetric
distribution, then that distribution must be bivariate standard Normal (which
can happen without the assumption of cross sample independence), and the
aforesaid pivotal convergence holds.
