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On the converse law of large numbers. (English) Zbl 1445.60031
Summary: Given a triangular array with \(m_n\) random variables in the \(n\)th row and a growth rate \(\{k_n\}_{n=1}^{\infty}\) with \(\limsup_{n\to\infty}(k_n/m_n)< 1\), 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.
MSC:
60F15 Strong limit theorems
28A35 Measures and integrals in product spaces
28E05 Nonstandard measure theory
03H15 Nonstandard models of arithmetic
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