Distribution-uniform anytime-valid sequential inference and the Robbins-Siegmund distributions

TL;DR

This work addresses the challenge of obtaining distribution- and time-uniform (i.e., $\mathcal{P}$-uniform and time-uniform) anytime-valid inference across a rich class of distributions $\mathcal{P}$ by introducing Robbins-Siegmund distributions as the limiting objects for suprema of transformed sums. It advances the theory by coupling strong Gaussian approximations with boundary-type limit results to craft distribution-uniform confidence sequences, anytime $p$-values, and sequential tests for means, under weak polynomial moment assumptions. A key contribution is the SeqGCM test for sequential conditional independence testing without Model-X, accompanied by a hardness result showing fundamental limits of unconditional sequential testing under $\mathcal{P}_0$. The framework yields robust, practical tools for sequential inference with modest moment conditions and opens avenues for extensions to martingale dependence, high-dimensional regimes, and Berry-Esseen-type refinements.

Abstract

This paper develops a theory of distribution- and time-uniform asymptotics, culminating in the first large-sample anytime-valid inference procedures that are shown to be uniformly valid in a rich class of distributions. Historically, anytime-valid methods -- including confidence sequences, anytime $p$-values, and sequential hypothesis tests -- have been justified nonasymptotically. By contrast, large-sample inference procedures such as those based on the central limit theorem occupy an important part of statistical toolbox due to their simplicity, universality, and the weak assumptions they make. While recent work has derived asymptotic analogues of anytime-valid methods, they were not distribution-uniform (also called \emph{honest}), meaning that their type-I errors may not be uniformly upper-bounded by the desired level in the limit. The theory and methods we outline resolve this tension, and they do so without imposing assumptions that are any stronger than the distribution-uniform fixed-$n$ (non-anytime-valid) counterparts or distribution-pointwise anytime-valid special cases. It is shown that certain ``Robbins-Siegmund'' probability distributions play roles in anytime-valid asymptotics analogous to those played by Gaussian distributions in standard asymptotics. As an application, we derive the first anytime-valid test of conditional independence without the Model-X assumption.

Figures (2)

• Figure 1: The left-hand side plot contains one- and two-sided confidence sequences for the mean of i.i.d. Uniform$[0,1]$ random variables. The right-hand side plot contains one- and two-sided anytime $p$-values for the null hypothesis of $\mathbb E_{\mathsf P}[X_1] = 1/2$ under the alternative $\mathbb E_{\mathsf P}[X_1] = 0.51$.
• Figure 2: Empirical cumulative type-I error rates and power for the fixed-$n$ GCM test of shah2020hardness versus the sequential GCM test (SeqGCM) in \ref{['theorem:seq-GCM']} with a target type-I error of $\alpha = 0.05$ in a simulated conditional independence testing problem.

Theorems & Definitions (54)

• Remark 1.1: On the terms "honesty" and "uniformity"
• Theorem 1.2: Distribution-uniform anytime-valid inference (brief & informal)
• Definition 2.1: $\mathcal{P}$-uniform strong consistency
• Proposition 2.2: $\mathcal{P}$-uniform almost-sure multiplicative consistency of the variance
• Definition 2.3
• Proposition 2.4: Calculus of $\widebar O_\mathcal{P}(\cdot)$ and $\widebar o_\mathcal{P}(\cdot)$
• Definition 2.5: The Robbins-Siegmund distributions
• Proposition 2.6
• Proposition 2.7: The Robbins-Siegmund distributions from transformed Gaussian processes
• Theorem 2.8: A simplification of Theorem 4 from waudby2025nonasymptotic