Hilbert space methods in the theory of time series

Abstract

The use of Hilbert space methods is practically universal amongst probabilists working in time series analysis. This thesis attempts to develop a completely general approach to these methods. It is based largely on the work of two authors, Parzen (1959,1961) and Rozanov (1963), who have both considered problems of a general nature. Parzen gives a good introduction to the representation of general second-order random functions, but he is concerned only with processes in one dimension, and there is a certain lack of rigour in some of his arguments. Rozanov gives a very sophisticated discussion of stationary processes in n dimensions, and offers probably the most extensive treatment of prediction theory to date. Hilbert space methods were first employed as a means of investigating the structure of time series by Karhunen (1947) and Loeve (1948). During this period, several fundamental theorems were proved, amongst them the spectral representation theorem for stationary processes, and the Karhunen-Loeve representation for random processes on a finite interval. These theorems have been used by several workers especially Grenander (1950), in problems of statistical inference on time series. One exception is a result, due to Loeve (1948), establishing an intimate connection between second-order random functions and reproducing-kernel Hilbert spaces. This found little application until it attracted the attention of Parzen. A general coverage of most of the results from the Karhunen-Loeve period, including a detailed discussion of Parzen's work, is given.