Weak Convergence: Introduction

Abstract

Up to now, we have concentrated on the convergence of {θ _n } or of {θⁿ(·)} to an appropriate limit set with probability one. In this chapter, we work with a weaker type of convergence. In practical applications, this weaker type of convergence most often yields exactly the same information about the asymptotic behavior as the probability one methods. Yet the methods of proof are simpler (indeed, often substantially simpler), and the conditions are weaker and more easily verifiable. The weak convergence methods have considerable advantages when dealing with complicated problems, such as those involving correlated noise, state dependent noise processes, decentralized or asynchronous algorithms, and discontinuities in the algorithm. If probability one convergence is still desired, starting with a weak convergence argument can allow one to “localize” the probability one proof, thereby simplifying both the argument and the conditions that are needed. For example, the weak convergence proof might tell us that the iterates spend the great bulk of the time very near some point. Then a “local” method such as that for the “linearized” algorithm in Theorem 6.1.2 can be used. The basic ideas have many applications to problems in process approximation and for getting limit theorems for sequences of random processes.