Probability Measures on Metric Spaces

In this book, the author gives a cohesive account of the theory of probability measures on complete metric spaces (which is viewed as an alternative approach to the general theory of stochastic processes). After a general description of the basics of topology on the set of measures, the author discusses regularity, tightness, and perfectness of measures, properties of sampling distributions, and metrizability and compactness theorems. Next, he describes arithmetic properties of probability measures on metric groups and locally compact abelian groups. Covered in detail are notions such as decomposability, infinite divisibility, idempotence, and their relevance to limit theorems for sums of infinitesimal random variables. The book concludes with numerous results related to limit theorems for probability measures on Hilbert spaces and on the space of continuous functions on an interval. This book is suitable for graduate students and researchers interested in probability and stochastic processes and would make an ideal supplementary reading or independent study text.

Parthasarathy builds far in advance of the general theory of stochastic processes as the theory of probability measures in complete separable metric spaces. He begins with the Borel subsets of a metric space and proceeds to explain probability measures in a metric space, probability measures in a metric group, probability measures in locally compact abelian groups, the Kolmogorov consistency theorem and conditional probability, probability measures in a Hilbert space, and probability measures on C[0,1] and D[0,1]. Parthasarathy includes a comprehensive bibliography and bibliographic notes. Annotation © 2006 Book News, Inc., Portland, OR