Essential correlatedness and almost independence
Two random variables \(X_ 1\) and \(X_ 2\) are said to be essentially correlated if there exist no Borel measurable functions \(f_ i\), \(i=1,2\), such that \(f_ 1(X_ 1)\) and \(f_ 2(X_ 2)\) are uncorrelated. Two random variables \(X_ 1\) and \(X_ 2\) are said to be almost independent if they are not independent but there exist almost identical Borel functions \(f_ 1\) and \(f_ 2\) such that \(f_ 1(X_ 1)\) is independent of \(X_ 2\) and \(f_ 2(X_ 2)\) is independent of \(X_ 1\) (A function \(f:R\to R\) is called almost identical if there exist distinct real numbers \(a\) and \(b\) such that \(f(x)=a\) if \(x\neq a\) and \(f(a)=b)\). The author proves the following theorem: For bounded random variables \(X_ 1\) and \(X_ 2\), \(X_ 1\) and \(X_ 2\) are essentially correlated iff they are almost independent. Related results on how to construct such pairs of random variables are discussed.
scientific article; zbMATH DE number 95738