Abstract
AbstractFor stochastic processes which are induced by a signed measure, the Andersen-Jessen theorem asserts almost sure convergence and yields the identification of the limit. This result has been extended to real and vector-valued stochastic processes which are induced by a finitely additive set function or a set function process. In the present paper, we study the structure of such induced stochastic processes in order to locate the Andersen-Jessen theorem and its extensions in the family of convergence theorems for martingales and their generalizations. As an application of these results, we also show that the Andersen-Jessen theorem and its extensions can be deduced from the convergence theorems for conditional expectations and positive supermartingales.
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