Weak convergence and the prediction process

We first consider convergence in law of measurable processes with a general parameter set and a state space. To this end, we need to investigate topological properties of the space of measurable functions which is the paths space of measurable processes. Also a characterization of compact sets in the space is derived and some functionals on the space are discussed.We then proceed to prove some properties of probability measures on the space of measurable functions. After investigating conditions on function spaces which guarantee that weak convergence may be proved by establishing finite-dimensional convergence and tightness, we prove necessary and sufficient conditions for convergence in law of measurable processes. These results are then applied to convergence of the prediction process; a special case where the given process is Markov is studied in detail.We also derive two results on the prediction process which is not in the field of weak convergence. One is on the asymptotic behavior of the prediction process under absolutely continuity, and the other is an example which shows that the prediction process may lost Markov property if one uses a larger filtration than the natural one.U of I OnlyETDs are only available to UIUC Users without author permissio