A Finite-Iteration Theory for Asynchronous Categorical Distributional Temporal-Difference Learning

We study finite-iteration behavior of the exact asynchronous recursions used by categorical distributional temporal-difference methods. The analysis covers scalar categorical TD in the Cramér geometry and multivariate signed-categorical TD in the maximum mean discrepancy geometry. Existing statewise isometric embeddings turn both methods into single-state stochastic-approximation recursions that contract in a block-supremum norm, but the categorical operators are contractive only on invariant representation domains. We establish the required restricted-domain theory and obtain discounted bounds under i.i.d. sampling and under a Markovian trajectory. A Poisson-equation decomposition handles trajectory dependence without an explicit mixing-time window. For undiscounted fixed-horizon policy evaluation, we establish analogous finite-iteration guarantees for horizon-stacked categorical methods under episodic sampling. Together, these results provide a unified non-asymptotic analysis of asynchronous categorical distributional TD across scalar, multivariate, discounted, and fixed-horizon settings.