On the converse law of large numbers

Abstract

Given a triangular array with mn random variables in the nth row and a growth rate {kn}n=1∞ with lim sup n→∞(kn/mn)<1, if the empirical distributions converge for any subarrays 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 nth 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 nth 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 subarrays 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.