Minimal symmetrization of random vectors in \(R^2\) (Q2737036)

It is known that a random variable \(\xi\) has a symmetric distribution if \(\xi\) and \(-\xi\) are identically distributed. If \(\xi_{1}\) and \(\xi_{2}\) are identically distributed (independence is not supposed), then the random variable \(\xi=\xi_{1}-\xi_{2}\) has a symmetric distribution. The aim of this paper is to extend the above-mentioned property to random vectors with values in \(n\)-measurable Euclidean space \(R^{n},\) \(n\geq 2.\) The properties of symmetrized vectors and analogues of the famous symmetric distributions are studied.