## Illinois Journal of Mathematics

### 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 $\limsup_{n\to\infty}(k_{n}/m_{n})\lt 1$, 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.

#### Article information

Source
Illinois J. Math., Volume 64, Number 2 (2020), 199-225.

Dates
Revised: 15 December 2019
First available in Project Euclid: 1 May 2020

https://projecteuclid.org/euclid.ijm/1588298628

Digital Object Identifier
doi:10.1215/00192082-8303485

Mathematical Reviews number (MathSciNet)
MR4092956

#### Citation

Keisler, H. Jerome; Sun, Yeneng. On the converse law of large numbers. Illinois J. Math. 64 (2020), no. 2, 199--225. doi:10.1215/00192082-8303485. https://projecteuclid.org/euclid.ijm/1588298628

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