  We present for the first time a supermartingale certificate for
$\omega$-regular specifications. We leverage the Robbins & Siegmund convergence
theorem to characterize supermartingale certificates for the almost-sure
acceptance of Streett conditions on general stochastic processes, which we call
Streett supermartingales. This enables effective verification and control of
discrete-time stochastic dynamical models with infinite state space under
$\omega$-regular and linear temporal logic specifications. Our result
generalises reachability, safety, reach-avoid, persistence and recurrence
specifications; our contribution applies to discrete-time stochastic dynamical
models and probabilistic programs with discrete and continuous state spaces and
distributions, and carries over to deterministic models and programs. We
provide a synthesis algorithm for control policies and Streett supermartingales
as proof certificates for $\omega$-regular objectives, which is sound and
complete for supermartingales and control policies with polynomial templates
and any stochastic dynamical model whose post-expectation is expressible as a
polynomial. We additionally provide an optimisation of our algorithm that
reduces the problem to satisfiability modulo theories, under the assumption
that templates and post-expectation are in piecewise linear form. We have built
a prototype and have demonstrated the efficacy of our approach on several
exemplar $\omega$-regular verification and control synthesis problems.
