The Tchebycheff statistical inequality and its generalizations are further generalized so as to apply equally well to $n$-dimensional probability distributions. Comparisons may be made with other generalizations ,  that have been developed recently for the two-dimensional case. The inequalities given in this paper are generally as close as the most favorable corresponding inequalities that exist for the one-dimensional case and in many simple cases they are closer than those that have been given heretofore for two dimensions. In a special case the upper bound of our inequality is actually attained. The theory contains also a less important generalization in one dimension.
"Generalization to $N$ Dimensions of Inequalities of the Tchebycheff Type." Ann. Math. Statist. 19 (4) 568 - 574, December, 1948. https://doi.org/10.1214/aoms/1177730152