The Pitman inequality for exchangeable random vectors

In this short article the following inequality called the “Pitman inequality” is proved for the exchangeable random vector (X1,X2,…,Xn)(X1,X2,…,Xn) without the assumption of continuity and symmetry for each component XiXi:P(|1n∑i=1nXi|≤|∑i=1nαiXi|)≥12   , where allαi≥0 are special weights with∑i=1nαi=1.