On the converse law of large numbers

Abstract

Given a triangular array with $m_n$ random variables in the $n$th row and a growth rate $\{k_n{\}}_{n=1}^{\infty }$ with ${lim\hspace{0.17em}sup\hspace{0.17em}}_{n\to \infty }(k_n/m_n)<1$, 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 $k_n$ 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 $k_n$ 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 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.