Moments of variables summing to gamma order statistics

Song, Buchberger, and Deddens (1992) derived means, variances, and covariances of variables summing to normal order statistics and proved that these variables are negatively correlated. In this paper we consider the case of variables summing to gamma order statistics. Formulas for the general product moments of these variables are obtained. Contrary to the normal case, variables summing to gamma order statistics are not necessarily negatively correlated. In addition, the covariances of variables corresponding to different order statistics are also obtained.