Diagonal Expansions of Bivariate Probability Distributions and their Application to Point Processes.

Abstract

The chief aim of this thesis is to study a class of bivariate distributions which have a diagonal representation in terms of orthogonal polynomials on the marginals, and their application to a particular class of point processes. In chapter one, we introduce the concept of such a class of bivariate distributions and briefly discuss some historical and more recent developments in this area. Wold's Markov process of intervals is also introduced. In the second chapter, problems in characterizing the sequences of canonical correlation coefficients are discussed. In particular, we characterize these coefficients in terms of moment sequences as well as positive definite sequences. We introduce the Meixner class of distributions in chapter three and show that it can be characterized in various ways. Meixner's hypergeometric distribution is studied in some detail. In chapter four, we prove in a unified way that is a bivariate distribution for all distributions (except Meixner's hypergeometric distribution) in the Meixner class. Characteristic functions corresponding to f(x,y) are also obtained. In chapter five, we devote our dicussion to Wold's point process which is a generalization of the ordinary renewal process. An integro-differential equation is derived and it is used to obtain the 'renewal equation'. Then the diagonal expansion theory for bivariate distributions is applied to obtain the spectrum of counts. Finally, in chapter six, we switch our discussion to the discrete analogue of wold's point process and establish a representation for the 'renewal function'. The spectrum of counts is also obtained.