Equivalence of Asymptotic Normality of the Two Sample Pivot and the Vector of Standardized Sample Means

From an infinite sequence of independent random vectors in the plane, where each coordinate sequence consists of identically distributed random variables that have a finite second moment, we construct a double sequence of random vectors consisting of the standardized sample means from the two coordinates with different sample sizes. We show that as the two sample sizes tend to infinity, convergence in distribution of this vector of standardized sample means to the standard Normal distribution on the plane, convergence in Cesaro means of the sequence of cross-sample correlation coefficients to 0, and convergence in distribution of the well-known two sample pivot for comparing the two coordinate means to the standard Normal distribution on the line are equivalent.