On the converse law of large numbers

Abstract

Given a triangular array with mn random variables in the n-th row and a growth rate {kn}n=1 with lim supn→∞(kn/mn) < 1, if the empirical distributions converge for any sub-arrays with the same growth rate, then the triangular array is asymptotically independent. In other words, if the empirical distribution of any kn random variables in the n-th row of the triangular array is asymptotically close in probability to the law of a randomly selected random variable among these kn random variables, then two randomly selected random variables from the n-th row of the triangular array are asymptotically close to being independent. This provides a converse law of large numbers by deriving asymptotic independence from a sample stability condition. It follows that a triangular array of random variables is asymptotically independent if and only if the empirical distributions converge for any sub-arrays with a given asymptotic density in (0, 1). Our proof is based on nonstandard analysis, a general method arisen from mathematical logic, and Loeb measure spaces in particular.