Weak Convergence for Unconstrained Systems

Abstract

In this chapter, we develop the weak convergence versions of most of the results of Chapter II. Since weak convergence methods are used, rather than the “w.p.1 convergence” methods of Chapter II, the conditions on the noise sequences are weaker than those used in Chapter II, but, in return, we no longer have convergence w.p.1. The convergence is, in many cases, stronger than convergence in probability, however. The general idea is similar to that in Chapter II, in that compactness methods (here based on weak convergence results) are used. The basic idea is to first prove tightness of {Xn(·)|, then to use the resulting compactness by extracting a weakly convergent subsequence and characterizing its limit. The limit processes satisfy the same ODE’s as in Chapter II, and, again, the properties of the ODE yield information on the asymptotic properties of {Xn|.