Necessary and Sufficient Condition for Asymptotic Normality of Standardized Sample Means

TL;DR

This paper establishes a precise condition involving cross sample correlation coefficients that determines when standardized sample means from an infinite sequence of independent, square-integrable random vectors are jointly asymptotically normal.

Contribution

It provides a necessary and sufficient condition for the asymptotic normality of standardized sample means based on the convergence of Cesaro means of cross correlation coefficients.

Findings

Asymptotic normality occurs if and only if Cesaro means of cross correlations converge to zero.

The result applies to infinite sequences of independent, square-integrable random vectors.

The condition links correlation structure to asymptotic distribution behavior.

Abstract

The double sequence of standardized sample means constructed from an infinite sequence of square integrable independent random vectors in the plane with identically distributed coordinates is jointly asymptotically Normal if and only if the Cesaro means of the sequence of cross sample correlation coefficients converges to 0.