Conditioned Limit Theorems

Limit theorems for Markoff processes and suitable functionals defined on the processes occur in two principal contexts. The first context treats a situation where the limit process is one of the classical stable processes. The usual approximating processes are sums of independent random variables. A second class of examples is that of a limit diffusion process of Bessel type. Then the approximating processes may themselves either be of diffusion type (i.e., random walks, birth and death or bona fide diffusion) or processes almost of diffusion type. In both cases under sufficient regularity conditions, there exists an invariance principle, i.e., the convergence of the processes entails the convergence in law of functionals continuous a.e. with respect to the limit process. The objective is to develop several limit laws for random variables subject to conditioning on a recurrent event.