Test Supermartingale Methods

• Test supermartingales are nonnegative stochastic processes with a nonincreasing conditional expectation, providing time-uniform and anytime-valid hypothesis tests.
• They are constructed by multiplying predictable factors that ensure the conditional expectation under the null hypothesis does not increase, thereby controlling the Type I error.
• This methodology underpins sequential testing, robust confidence intervals, safety certifications, and algorithmic randomness assessments in both discrete and continuous settings.

A test supermartingale is a nonnegative stochastic process, adapted to a filtration, whose conditional expectation is nonincreasing under a specified null hypothesis, and which provides rigorously time-uniform certificates for statistical testing, probabilistic inference, and safety verification. This formalism underlies a broad array of modern sequential test designs, robust confidence intervals, algorithmic randomness frameworks, and certification tools for stochastic dynamical systems. Large values of test supermartingales indicate evidence against the null hypothesis, and their construction guarantees control of Type I error uniformly over arbitrary stopping rules and composite nulls.

1. Definition and Fundamental Properties

Let $(X_n)_{n\ge0}$ be a stochastic process adapted to filtration $(\mathcal F_n)_{n\ge0}$ and let $H_0$ denote a null hypothesis about the law of the underlying data. A test supermartingale $M_n$ for $H_0$ is constructed such that:

• $M_0=1$,
• $M_n\ge0$ and $\mathcal F_n$-measurable,
• Under every law in $H_0$, for all $n$, $$E_{H_0}\left[M_{n+1} \mid \mathcal F_n \right] \le M_n$$.

This property ensures $M_n$ is a nonnegative supermartingale under any law conforming to $H_0$ (Wills et al., 2017, Hendriks, 2021, Hendriks, 2018).

Ville’s inequality applies: $$P_{H_0}\left(\exists\,n : M_n \ge 1/\alpha\right) \le \alpha\,,$$ for any $\alpha\in(0,1)$, providing an anytime-valid level-$\alpha$ test via thresholding. This result generalizes both fixed-sample p-value bounds and sequential test properties (Wills et al., 2017).

2. Construction Methodologies

Test supermartingales are typically constructed by multiplying predictable factors whose conditional expectation under $H_0$ does not exceed $1$, ensuring the supermartingale property. For bounded random variables $X_i \in [a,b]$ and $H_0: E[X_i] \le \mu$, an explicit family is

$$M_n = \prod_{i=1}^n \left(1 + \lambda_i (X_i - \mu)\right)\,,$$

with each $\lambda_i$ chosen in the admissible range determined by $[a,b]$ and filtration $\mathcal F_{i-1}$ (Hendriks, 2021, Hendriks, 2018).

Adaptive strategies include optimizing or randomizing $\lambda_i$ based on observed data, and mixture constructions over families of parameters. Integrated test supermartingales, using priors over allowed parameters, enhance robustness and power (Hendriks, 2021, Hendriks, 2018).

For composite or multidimensional nulls, intersection supermartingale products across strata (ALPHA-type constructions) and union-intersection p-value maximizations are used for risk-limiting audits and other stratified inference problems (Spertus et al., 2022).

In continuous-time stochastic systems, the test supermartingale concept is extended to Itô SDEs via infinitesimal generator analysis: $$L V(x) = \nabla V(x)^\top f(x) + \frac{1}{2} \operatorname{Tr}[G(x)^\top \nabla^2 V(x) G(x)]\,,$$ and certificates satisfying the local decrease condition $L V(x) \le -\alpha(x)$ provide probabilistic safety guarantees via supermartingale arguments (Neustroev et al., 2024).

3. Statistical Testing, Confidence Intervals, and Sequential Inference

Test supermartingales underlie sequential hypothesis tests: rejection occurs at the first time $n$ with $M_n \ge 1/\alpha$, with Type I error control at level $\alpha$, valid for any stopping rule (Hendriks, 2021, Hendriks, 2018, Wills et al., 2017).

By inversion, one obtains time-uniform confidence bounds. For bounded data and null $E[X] \le \mu$, the lower confidence bound is

$$L_n(\alpha) = \sup\left\{\mu : \sup_{k \le n} M_k^\mu \le 1/\alpha\right\}\,,$$

valid as $P(L_n(\alpha) \le E[X]) \ge 1-\alpha$ for all $n$ (Hendriks, 2021). Two-sided intervals follow by combining one-sided supermartingales.

The methodology has been specialized to Bernoulli trials, yielding explicit closed-form p-value bounds and confidence intervals, with quantified comparative performance to classical fixed-sample bounds (exact binomial, Chernoff–Hoeffding) (Wills et al., 2017). The primary cost of full stopping-rule robustness is a mild widening of intervals ($\sim \frac{1}{2}\ln n$) compared to exact tests (Wills et al., 2017).

For statistical models with monotone likelihood ratios (MLR) and sufficient statistics, sequential likelihood-ratio processes constructed via those statistics are shown to be test supermartingales, enabling rigorous sequential t-tests, $\chi^2$-tests, and regression with nuisance covariates (Grünwald et al., 6 Feb 2025).

4. Algorithmic Randomness and Universal Tests

In the context of sequence randomness and computable forecasting systems, test supermartingales characterize Martin–Löf test randomness: a path is random if no lower-semicomputable test supermartingale diverges along it. Universal Martin–Löf tests and universal test supermartingales are constructible, dominating all lower-semicomputable test supermartingales (Cooman et al., 2023).

Randomness notions such as Schnorr randomness are equivalently characterized via computable test supermartingales; failures correspond to explosive growth of the supermartingale along certain paths (Cooman et al., 2023).

5. Extensions: Continuous-Time, Neural Certification, and Strong Supermartingales

Continuous-time extensions embed the test supermartingale condition in the trajectory analysis of Itô diffusions, where neural network parameterizations $V_\theta(x)$ enforce the supermartingale decrease condition across domains, utilized for safety verification in physical systems (Neustroev et al., 2024). Training leverages automatic differentiation and interval-bound propagation to guarantee formal certificates on reach-avoid and persistence specifications.

Strong supermartingales and Snell envelopes in optimal stopping theory are linked as minimal dominating supermartingales, characterized via Mertens decomposition and the Skorokhod minimal-push property. These concepts generalize test supermartingale arguments for reflected BSDEs and stochastic control, where the minimal envelope test ensures optimality and minimality (Aazizi et al., 2011).

6. Quantitative Rate Results and Relaxed Conditions

Recent work quantifies the convergence of sequences with relaxed supermartingale assumptions, providing explicit rates for mean and almost sure convergence under uniform integrability and controlled additive perturbations. A general theorem yields rates dependent only on easily computed functionals of the error terms and “slow-down” modulus, which are extensible to stochastic approximation, Dvoretzky-type convergence, quasi-Fejér monotonicity, and fast-rate Robbins-Siegmund settings (Neri et al., 17 Apr 2025).

Linear and sublinear convergence rates are recoverable under standard conditions, directly connecting test supermartingale methodology with the theory of iterative stochastic algorithms (Neri et al., 17 Apr 2025).

7. Applications and Impact

Test supermartingales are widely applied across domains:

• Statistical sequential hypothesis tests with anytime-valid error control (Hendriks, 2021, Wills et al., 2017, Grünwald et al., 6 Feb 2025).
• Time-uniform confidence intervals and adaptive inference (Wills et al., 2017, Hendriks, 2021).
• Risk-limiting audits and stratified sampling with union–intersection supermartingale products (Spertus et al., 2022).
• Formal certification for nonlinear stochastic systems safety using neural architectures (Neustroev et al., 2024).
• Foundations of algorithmic randomness with universal tests (Cooman et al., 2023).
• Stochastic optimization and stochastic approximation with effectively computable rates (Neri et al., 17 Apr 2025).
• Strong envelope constructions for reflected BSDEs and optimal stopping (Aazizi et al., 2011).

Test supermartingale methodology consolidates key advances in sequential analysis, robust inference, probabilistic certification, and theoretical computer science, enabling principled statistical reasoning under strong distributional uncertainty and arbitrary data-dependent stopping rules.