Nonnegative test supermartingales

Assume the Axiom of Choice (AC) The Axiom of Choice. Let $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\in {\mathbb{N}}_0})$ be a discrete filtration on a measurable space, and let ${\mathcal{P}}_0$ be a family of probability laws on $(\mathrm{\Omega },\mathcal{F})$. A real-valued process $E=(E_t{)}_{t\in {\mathbb{N}}_0}$ is a nonnegative test supermartingale for ${\mathcal{P}}_0$ if, for every $P\in {\mathcal{P}}_0$, it is adapted and integrable under $P$ Filtration and filtered probability space Martingale submartingale and supermartingale, satisfies $P(E_t\ge 0)=1$ for every $t$, and has ${\mathbb{E}}_P{E}_0\le 1$.

The same process must satisfy the supermartingale condition under every law in the null family; that is what makes the definition uniform over a composite null. The initial value may be random and need not be bounded by one pointwise. Time starts at zero.

A nonnegative test supermartingale is a standard sufficient process for anytime-valid testing. Some literature uses e-process for a broader class, allowing a process dominated under each null law by a different test martingale. This item defines the planned common test-supermartingale class; it does not assert that every e-process is itself a supermartingale.