SOME PROPERTIES OF DISTRIBUTION FUNCTIONS AND TRANSFORMATIONS THAT INDUCE ONE ANOTHER.

Abstract : This report contains results of research on families of transformations of random variables and families of distribution functions. Each of the families of transformations is such that it induces a fixed distribution function when a given random variable is subjected to any member of the family. Necessary and sufficient conditions are given under which a given family is a non-empty set and in particular it is shown that these depend only on the saltuses of the induced distribution function and of the distribution function corresponding to the random variable being transformed. Each of the families of distribution functions is such that it induces a fixed distribution function when random variables associated with the family are subjected to a fixed transformation. Each family is treated as a convex subset of the function space whose points are Baire functions. Properties of the function space itself are developed in some detail and particular attention is given to questions of the existence of boundary points of the convex subsets. Other related topics are also discussed. (Author)