SOME MULTIVARIATE CHEBYSHEV INEQUALITIES WITH EXTENSIONS TO CONTINUOUS PARAMETER PROCESSES

0. Summary. 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. 1. Introduction. We begin by discussing a model for the various generalizations of Chebyshev's inequality, and for a standard proof that we shall use. Examination of this proof will enable us to make some general comments concerning the problems of deriving inequalities and of proving sharpness. Let (Q, B, P) be a probability space, and let ($C, (1) be a measurable space. For each i c I, an arbitrary index set, let Bi C (t and let 5i be a class of random