The Large Deviations of random time - changesbyRaymond

This thesis is concerned with the transformation of the Large Deviation properties of a stochastic process under a random time-change. A random time-change is the reparametrisation of a process by a lower adjoint process; a simple example is the sequence of random variables fSTn gn2N constructed by stopping a process fSt gt2R+ at a sequence of random times f Tn gn2N . Such a sequence of random times is lower adjoint to its associated point process fNt gt2R+ , de ned byNt := sup fn 2 N : Tn 6 t g : In order to study the Large Deviations of random time-changes, we need to understand how the Large Deviation behaviour of adjoint processes are related. We prove that, under simple and very general hypotheses, an upper adjoint process satis es a one-dimensional Large Deviation principle with rate-function U if and only if its lower adjoint does with rate-function V , where V (y) = y U(1=y): The proof of this result is almost purely one-dimensional; it does, however, involve a weak mixing condition. A similar one-dimensional approach to the case of random time-changes would require even stronger mixing conditions and would be somewhat awkward. Since a random time-change is a transformation of sample-paths, and the mixing conditions required to establish the relationship between the Large Deviations of a process and a random timechange of it are most conveniently stated in terms of sample-paths, it is most natural to consider the sample-path Large Deviations of adjoint processes and random time-changes. Adjoint processes are probability measures on the space of Galois connections between two partially ordered sets. We present the properties of Galois connections in general, and show how to construct the space of all adjunctions between I and J , where I and J?, the dual of J , are continuous Heyting algebras. We study the order-theoretic properties of the random time-change transformation and prove that it is a homeomorphism in the order topology. This is the key result, as far as Large Deviations is concerned, for it allows us to apply the Contraction Principle and deduce that a process satis es a sample-path Large Deviation principle jointly with an upper adjoint process if and only if the random time-change does. We investigate the relationship of the sample-path Large Deviation behaviour of adjoint processes and random time-changes with their one-dimensional Large Deviations. We present the random time-change formula, which relates the joint one-dimensional rate-function U of a process fSj g and an upper adjoint process fXj g with the joint rate-function V of the random time-change SYi and the lower adjoint fYi g: V (s; y) = y U(s=y; 1=y): We also investigate the relationship between the order topology on sample-path space with the other intrinsic lattice topologies and prove that, in the lattice of Galois connections between two complete chains, all the classic intrinsic lattice topologies are homeomorphic. Acknowledgements I found this part of my thesis the most challenging to write. There are so many people who have helped so much with the work which I present here that I cannot hope to express fully my thanks; all I can do is acknowledge the main contributors. John Lewis has been my supervisor, mentor and friend for over four years, and it would have been impossible for me to have done this work without him. I know from personal experience and from talking to many of my friends how very lucky I am to have John as a supervisor. He has always been tremendously supportive in every way, consistently full of enthusiasm for my work and ready to discuss my ideas with me at any moment. He has worked hard to create a seemingly endless sequence of new opportunities for me over the years and I have bene ted enormously from them. Working with John has been one of the most enjoyable and ful lling experiences of my life and I am deeply grateful for everything he has done for me. It has been my privilege to work with a dynamic group of fascinating people; for all that I have learned from them and all the excitement I have shared with them, my warmest thanks go to all the members, past and present, of the Applied Probability Group, especially my colleague and friend Fergal Toomey. The sta of the Dublin Institute for Advanced Studies strive hard to provide a warm and stimulating environment; the resources which they provide have been instrumental in helping me write this thesis. My heartfelt thanks go to them all, especially Ann Goldsmith and Margaret Matthews, for their support. Prof. David Simms of the School of Mathematics has ensured the smoothness and ease of all my dealings with Trinity College Dublin. I am grateful to him for taking the time and e ort to act as my internal supervisor. I owe a great debt to all my teachers over the years | far greater, I am sure, than I am aware of. My lecturers in Trinity College gave me a fascinating four years with an excellent course in mathematics and physics. I cannot begin to explain what my primary and secondary school teachers have given me; any successes I may have achieved are re ections of their skill. I would like to mention in particular Mr. John Brown, whose boundless enthusiasm for mathematics is, more than anything, what led me to study the subject. Above all others, my parents have contributed to this thesis in so many and such fundamental ways. They brought me into this world, cared for me in ways I have only recently begun to appreciate clearly, and gave me a wonderful upbringing and education. I could not ask for better parents and appreciate deeply their love and support. Whenever I tell anyone that I have three older sisters, they always say that I must have been spoiled rotten. I try to deny it but it is, of course, true. My sisters are simply the best and I appreciate all the understanding and love they have given me. Adrienne has brought a huge amount of love and happiness into my life. She has helped me so much with my work, paradoxically, by dragging me away from it, up and down mountains and out onto the water. She has been wonderfully patient and supportive, keeping me laughing and only once recently reminding me that, ever since I met her, I have been claiming to be \nearly nished my thesis." I have many good friends who have given me endless encouragement and support; their friendship means the world to me. I thank them all, especially John Monahan, the best friend anyone could have, for keeping me sane. On the technical side, my thanks go to Neil O'Connell for providing much of the inspiration for the ideas in this thesis. My thanks go also to John Walsh for putting the Compendium in my hands; had he not done so, my thesis would have been very di erent. I have a dedicated team of proof-readers, Ken Du y, John Lewis, Brian McGurk and Meriel Huggard, led by the incomparable Cormac Walsh. I am very grateful to them for the e orts they made and all the errors they caught in the very short time I allowed them. Thanks also to Donald Knuth, Leslie Lamport and everyone else on the TEX/LATEX team for their beautiful typesetting system. To everyone, thank you.