Conditional Probability Operators

Abstract

The study of laws of random variables is facilitated by various methods of representing these laws. The distribution function and characteristic function have played an important role, and the functional representation of the law of a random variable $X$ as a mapping $T$ of the bounded Borel functions into the real line given by $Tg = Eg(X)$ connects many problems in probability theory with functional analysis. Similarly, in the study of conditional probability laws various representations of these conditional laws are desirable. In particular, in the study of Markov processes an operator representation has proven most useful. In this paper we develop the representation of the conditional law of a random variable in a way analogous to the functional representation mentioned above. Thus, if $X$ is a random variable on a probability space $(\Omega, \mathcal{A}, P)$ and $\mathcal{A}^\ast$ is a sub-$\sigma$-field of $\mathcal{A}$, then we introduce the $\mathcal{A}^\ast$ conditional operator $T$ of $X$ mapping the bounded Borel functions into $L_\infty(\Omega, \mathcal{A}^\ast, P)$ and given by $T_g = E^{\mathcal{A}\ast} g(X)$. The first four sections of the paper develop a rudimentary theory of such operators. The use of these operators in probability theory leads one to consider various operator topologies, etc., weaker than those usually studied in functional analysis. Most of the material presented in this development is quite elementary, and many properties of these operators having potential interest have not even been mentioned. Nevertheless, it is hoped that this exposition may suggest further use of the operator representation of conditional distributions in probability theory, and that the properties of these operators relevant to probability theory will be investigated more systematically. In the final section conditional probability operators are applied to a mixing problem. A stationary process is said to have central structure if, under conditions similar to those of the central limit problem for independent random variables, conditional limit laws given the invariant $\sigma$-field are infinitely divisible. It is shown that central structure is closely related to a type of uniform ergodicity.