Probability Theory- A Foundational Course

This book shares the dictum of J. L. Doob in treating Probability Theory as a branch of Measure Theory and establishes this relation early. Probability measures in product spaces are introduced right at the start by way of laying the ground work to later claim the existence of stochastic processes with prescribed finite dimensional distributions. Other topics analyzed in the book include supports of probability measures, zero-one laws in product measure spaces, Erdös-Kac invariance principle, functional central limit theorem and functional law of the iterated logarithm for independent variables, Skorohod embedding, and the use of analytic functions of a complex variable in the study of geometric ergodicity in Markov chains.

This book is offered as a text book for students pursuing graduate programs in Mathematics and or Statistics. The book aims to help the teacher present the theory with ease, and to help the student sustain his interest and joy in learning the subject.

Contents

1 Probability Measures in Product Spaces
2 Weak Convergence of Probability Measures
3 Characteristic Functions
4 Independence
5 The Central Limit Theorem and its Ramifications
6 The law of the iterated logarithm
7 Discrete Time Markov Chains
Index