\(\epsilon\)-dependence of \(X_ 1+X_ 2\) and \(X_ 1-X_ 2\)
According to a famous theorem of Bernstein; if \(X_ 1\) and \(X_ 2\) are independent, then \(X_ 1+X_ 2\) and \(X_ 1-X_ 2\) are independent if and only if \(X_ 1\) and \(X_ 2\) are normal random variables. In this paper the following interesting extension is given. Define the weak metric of the r.v.'s namely \[ \lambda (X,Y)=\lambda (f_ X,f_ Y)=\inf \{\max \{\nu (X,Y;t),t\}:t>0\}, \] where \[ \nu (X,Y;t)=(1/2)\max \{| f_ X(u)-f_ Y(u)|:| u| <1/t\},\quad f_ X(u)=E \exp \{i(u,X)\}. \] Call X and Y (\(\lambda\),\(\epsilon)\)-independent if \(\lambda (f_{(X,Y)},f_ X\cdot f_ Y)\leq \epsilon.\) Theorem: If \(X_ 1\) and \(X_ 2\) are independent and \(X_ 1+X_ 2\) and \(X_ 1-X_ 2\) are (\(\lambda\),\(\epsilon)\)-independent then \[ (1)\quad \inf_{\eta_ j\in \Phi}\lambda (X_ j,\eta_ j)\leq C\cdot \epsilon \quad j=1,2 \] where C is an absolute constant and \(\Phi\) is the class of normal random variables. Moreover the order in (1) is exact.
characterization of the normal distribution
scientific article; zbMATH DE number 3982248