ABSTRACT
CONTENTS 1.1 Some Concepts of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Independent Random Variables, Martingales, and Martingale
Difference Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Markov Chains with State Space (Rm,Bm) . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Mixing Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
In the convergence analysis of recursive estimation algorithms the noise properties play a crucial role. Depending on the problems under consideration, the noise may be with various properties such as mutually independent random vectors, martingales, martingale difference sequences, Markov chains, mixing sequences, stationary processes, etc. The noise may also be composed of a combination of such kind of processes. In order to understand the convergence analysis to be presented in the coming chapters, the properties of above-mentioned random sequences are described here. Without any attempt to give a complete theory, we restrict ourselves to present the theory at the level that is necessary for reading the book.
