Bounds for distribution functions of sums of squares and radial errors (Q1178799)

The authors give the upper and lower pointwise bounds for distribution functions of \(X^ 2+Y^ 2\) and \((X^ 2+Y^ 2)^{1/2}\) where \(X\) and \(Y\) are arbitrary random variables. These bounds are best possible in a sense. To be more precisely, fix marginals of \(X\) and \(Y\) and assume that they are symmetrical about zero. For any \(t\in R_ 1\) there exist \(X\), \(Y\) such that the value \(P(X^ 2+Y^ 2<t)\) coincides with the upper (lower) bound. A special attention is given to the case of normal random vector.