Abstract nonlinear prediction and operator martingales

In the first part of this paper, nonlinear prediction theory of vector valued random variables in Orlicz spaces is presented. The spaces need not be reflexive and the results of this part are essentially best possible for these spaces. The second part considers operator valued martingales in the strong operator topology and various convergence theorems are proved for them. Again the results are optimal for the Orlicz space situation. These are specialized to the scalar case showing that the well-known martingale convergence theorem can be obtained from the well-known Andersen-Jessen theorem. A few applications are also given. The same ideas and methods of computation unify the otherwise almost independent parts.