The Frequency Interpretation in Probability

Abstract

We formulate and discuss the notion of generic sequence (and random sequence) associated with a sequence of random variables. These are substantial generalizations of the notion of "collective" introduced by R. von Mises as an attempt to give an operational meaning to various probabilistic ideas. The definition of collectives lacked a precise formulation, and A. Church and others attempted to give a rigorous meaning to the theory by utilizing ideas from the theory of computability. Nevertheless, the theory had major flaws. Even though some important contributions to special cases were made by P. Martin-Löf, the whole circle of ideas has languished due to lack of generality. The theory presented in the present article resolves many of these problems, and provides a coherent framework for the relevant ideas. © 1999 Academic Press.