In this paper we obtain some multivariate generalizations of Chebyshev's inequality, two of which are extended to continuous parameter stochastic processes. The extensions are obtained in a natural way by taking into account separability and letting the number of variables approach infinity. Particular attention is paid to the question of sharpness. To show that the bound of the inequality cannot be improved, examples are given in a number of cases that attain equality.
"Some Multivariate Chebyshev Inequalities with Extensions to Continuous Parameter Processes." Ann. Math. Statist. 32 (3) 687 - 703, September, 1961. https://doi.org/10.1214/aoms/1177704964