Convergence of martingales via enriched dagger categories
Paolo Perrone, Ruben Van Belle
TL;DR
The paper develops a novel framework that uses topologically enriched dagger categories to study martingale convergence. By modeling probability spaces, Markov kernels, and random variables categorically, and by enriching the categories to capture pointwise convergence, it proves mean convergence results for martingales and backward martingales indexed by arbitrary nets, including vector-valued (Bochner) cases. Central to the approach is the Levi property for idempotents, which connects order limits of filtrations to topological limits via enriched functors. The work also establishes that idempotent kernels correspond to sub-sigma-algebras, and that conditional expectations arise as dagger-split idempotents, enabling a clean categorical formulation of filtrations and martingales. These results extend to vector-valued settings and yield new convergence theorems, suggesting broad applicability of topological enrichment in analysis and probability.
Abstract
We provide a categorical proof of convergence for martingales and backward martingales in mean, using enriched category theory. The enrichment we use is in topological spaces, with their canonical closed monoidal structure, which encodes a version of pointwise convergence. We work in a topologically enriched dagger category of probability spaces and Markov kernels up to almost sure equality. In this category we can describe conditional expectations exactly as dagger-split idempotent morphisms, and filtrations can be encoded as directed nets of split idempotents, with their canonical partial order structure. As we show, every increasing (or decreasing) net of idempotents tends topologically to its supremum (or infimum). Random variables on a probability space form contravariant functors into categories of Hilbert and Banach spaces, which we can enrich topologically using the L^p norms. Martingales and backward martingales can be defined in terms of these functors. Since enriched functors preserve convergence of nets, we obtain convergence in the L^p norms. The convergence result for backward martingales indexed by an arbitrary net, in particular, seems to be new. By changing the functor, one can describe more general notions of conditional expectations and martingales, and if the functor is enriched, one automatically obtains a convergence result. For instance, one can recover the Bochner-based notion of vector-valued conditional expectation, and the convergence of martingales with values in an arbitrary Banach space. This work seems to be the first application of topologically enriched categories to analysis and probability in the literature. We hope that this enrichment, so often overlooked in the past, will be used in the future to obtain further convergence results categorically.