Maximal Independent Stochastic Processes

Abstract

This paper concerns the following problem posed by R. Pyke (1958). What is the cardinality, $M_k$, of the maximal family of stochastically independent random variables defined on a given space $\Omega$, of cardinality ${\bar{\bar{\Omega}}} = k$? Since maximality is sought, the investigation is limited to two-valued, nontrivial (tvnt) random variables; and the $\sigma$-algebra of measurable subsets of $\Omega$ is taken to be that generated by the family of random variables. With these restrictions the problem is essentially one of cardinality. The results are summarized in the table below. Theorem 1 follows from the fact that stochastic independence entails the non-vanishing of certain finite intersections of elementary sets. Theorem 2 is a result of Kakutani, Kodaira and Oxtoby [4, 5, 6]. Theorem 3 is a consequence of a set theoretic result of Tarski [11], and a theorem of Banach [1, 2, 10, 11], which results were used in proofs of Theorem 2. Theorem 4 follows from a construction and a lemma of Marczewski [9]. The paper is divided into five sections. Section 1 introduces the notation and terminology. Section 2 discusses two types of independence. Sections 3, 4, and 5 treat, respectively, the finite, non-countable and countable cases.