The theory and application of stochastic processes

The work presented in this thesis was done during the period October, 1953 to July, 1955. The work is original in the sense that it was wholly done by myself. However, in the course of presenting the results it has been necessary to refer to the work of other men. Where this has been done it has always been made clear by means of a reference. Chapter I contains no original research but is rather a review of the existing state of the subject. In #2.2. a theorem due to Pitman is proved so that the only original research in Chapter 2 is contained in # 2.3. where Pitman’s theorem is extended. Chapter 3 is based on a paper by Ogawara, but most of the chapter is original and consists either of extensions of Ogawara’s results or of an examination of their efficiency. Chapter 4, 5 and 6 present original results obtained by the author, while Chapter 7, where a theory due to Dr. E.G. Bowen is tested, is also wholly original. Most of the contents of Chapter 3 and 4 were published by the author in a paper in Biometrika, Volume 42, parts 1 and 2. Most of the contents of Chapter 5 will appear under the author’s name in Biometrika, Volume 42, Parts 3 and 4. The material making up Chapter 7 appeared as an article by the author in The Australian Journal of Physics, Volume 8, No. 2. Most of the subject matter of Chapter 6 has been communicated (by Professor P.A.P. Moran) for publication under the author’s name in The Proceedings of the Cambridge Philosophical Society