On a General Concept of "In Probability"

Abstract

Chernoff [1] has called attention to a paper of Mann and Wald [5], which provides a general theory and a convenient notation for the derivation of theorems concerning stochastic limits and limit distributions. The present paper attempts to clarify the first of these topics, stochastic limits, by applying one form of the definition of convergence in probability to any event rather than just to convergence. As in the convergence case, most of the reasoning one is intuitively disposed to do in this connection is valid. Its justification is made more transparent but no more difficult by broadening the applicability of the definition. Thus broadened, "in probability" neither implies nor follows from "with probability one."