Convergence with Probability One: Correlated Noise

Abstract

The most powerful method in Chapter 5 was the “general compactness method” of Section 5.3. The basic assumption for that method is that the asymptotic “rate of change” of the martingale M ⁰(·) is zero with probability one. It was shown in Subsection 5.3.2 that the rate of change condition is quite weak and yields convergence under what are probably close to the weakest conditions on both the noise and step size sequences, when the underlying noise is of the martingale difference form. In this chapter it is shown that the same approach yields excellent results for correlated noise. General assumptions of the type used in Section 5.3 are stated in Subsection 1.1 and the main convergence theorem is proved in a straightforward way in Subsection 1.2.