SpringerBriefs in Probability and Mathematical Statistics

This book deals with topics in the area of Levy processes and infinitely divisible distributions such as Ornstein-Uhlenbeck type processes, selfsimilar additive processes and multivariate subordination. These topics are developed around a decreasing chain of classes of distributions Lm, m = 0,1,...,infinity, from the class L0 of selfdecomposable distributions to the class Linfinity generated by stable distributions through convolution and convergence.The book is divided into five chapters. Chapter 1 studies basic properties of Lm classes needed for the subsequent chapters.  Chapter 2 introduces Ornstein-Uhlenbeck type processes generated by a Levy process through stochastic integrals based on Levy processes. Necessary and sufficient conditions are given for a generating Levy process so that the OU type process has a limit distribution of Lm class.Chapter 3 establishes the correspondence between selfsimilar additive processes and selfdecomposable distributions and makes a close inspection of the Lamperti transformation, which transforms selfsimilar additive processes and stationary type OU processes to each other.  Chapter 4 studies multivariate subordination of a cone-parameter Levy process by a cone-valued Levy process.  Finally, Chapter 5 studies strictly stable and Lm properties inherited by the subordinated process in multivariate subordination.In this revised edition, new material is included on advances in these topics. It is rewritten as self-contained as possible. Theorems, lemmas, propositions, examples and remarks were reorganized; some were deleted and others were newly added. The historical notes at the end of each chapter were enlarged.This book is addressed to graduate students and researchers in probability and mathematical statistics who are interested in learning more on Levy processes and infinitely divisible distributions.