Composite Ville's Theorem Overview

• Composite Ville's Theorem is a nonparametric extension that unifies classical martingale results across a family of probability measures using e-processes.
• It characterizes composite null sets by the divergence of e-processes, ensuring robust sequential testing and inference under uncertainty.
• The theorem underpins key applications such as strong laws, minimax duality in game-theoretic probability, and advances in online learning.

Composite Ville's Theorem provides a fundamental extension of Ville's classical martingale theorem, enabling measure-theoretic and game-theoretic probability to be unified under a flexible, nonparametric framework. Rather than working under a single probability measure, composite Ville's theorem applies to arbitrary families of probability measures and utilizes "e-processes" as the correct generalization of nonnegative martingales. In this setting, events of outer measure zero—composite nullsets—are characterized by the existence of an e-process which explodes to infinity precisely when those events occur. This duality underlies robust sequential inference, strong laws, and minimax theorems in game-theoretic probability frameworks (Ruf et al., 2022, Frongillo, 24 Dec 2025).

1. Foundations and Definitions

Composite Ville’s theorem is set in the context of a filtered measurable space $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge0})$, with $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$. Instead of fixing a single law, one prescribes a (possibly large, nonparametric) collection $\mathcal P$ of probability measures on $(\Omega,\mathcal F)$. The set of all $\mathcal F_t$–stopping times is denoted $\mathcal T$.

Two key constructs generalize classical tools:

• Inverse Capital Outer Measure: For $A\subseteq\Omega$,

$$\mu^*(A) := \inf\left\{ \sup_{P\in\mathcal P} P(\tau<\infty) : A\subseteq\{\tau<\infty\},\; \tau\in\mathcal T \right\} \in [0,1].$$

$\mu^*$ is an outer measure; $A$ is a composite nullset if $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$0.

• $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$1–e-Process: A nonnegative, $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$2–adapted process $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$3 with $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$4 such that for every stopping time $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$5,

$$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$6

For every $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$7, there exists a nonnegative $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$8–martingale $$\mathcal F = \sigma\bigl(\bigcup_{t\ge0}\mathcal F_t\bigr)$$9 with $\mathcal P$0 almost surely for all $\mathcal P$1.

2. Formal Statement and Proof Outline

The composite Ville’s theorem provides necessary and sufficient conditions for an event being composite null and the existence of an e-process diverging on that event:

$\mathcal P$2

More generally, for $\mathcal P$3,

$\mathcal P$4

Outline of proof:

• If $\mathcal P$5: For each $\mathcal P$6, exist $\mathcal P$7 so that $\mathcal P$8 and $\mathcal P$9. Construct $(\Omega,\mathcal F)$0; this diverges on $(\Omega,\mathcal F)$1 and satisfies the e-process expectation bound.
• Only if $(\Omega,\mathcal F)$2 on $(\Omega,\mathcal F)$3: For $(\Omega,\mathcal F)$4, let $(\Omega,\mathcal F)$5. On $(\Omega,\mathcal F)$6, $(\Omega,\mathcal F)$7. Using the e-process property, $(\Omega,\mathcal F)$8. As $(\Omega,\mathcal F)$9, this ensures $\mathcal F_t$0 (Ruf et al., 2022).

3. Game-Theoretic Probability and Minimax Duality

Composite Ville's theorem can be interpreted in two-player game-theoretic probability, where the Gambler selects strategies and the World selects outcome paths or probability measures. Three "prices" are relevant:

Price	Definition	Context
Game-theoretic (Gambler first)	$\mathcal F_t$1	betting replication
Measure-theoretic (World first)	$\mathcal F_t$2	probabilistic
Composite (worst-case P)	$\mathcal F_t$3	robust consistency

Minimax duality for sequential gambling states:

$\mathcal F_t$4

where $\mathcal F_t$5 is the cumulative payoff of a strategy $\mathcal F_t$6. This provides the foundation for supermartingale constructions that diverge on composite-null events (Frongillo, 24 Dec 2025).

4. Comparison with Classical Ville’s Theorem

Composite Ville subsumes the classical theorem:

• For singleton $\mathcal F_t$7, $\mathcal F_t$8 and $\mathcal F_t$9–e–processes are precisely $\mathcal T$0–martingales. The theorem reduces to: $\mathcal T$1 iff exists nonnegative martingale $\mathcal T$2 on $\mathcal T$3.
• For general $\mathcal T$4, inverse capital measure replaces probability, and e-processes replace martingales. Stopping times become composite tests with uniform type-I error control across $\mathcal T$5 (Ruf et al., 2022).

5. Applications and Illustrative Examples

Composite Ville’s theorem underpins robust statistical inference and testing in nonparametric settings:

• Composite Strong Law of Large Numbers: For $\mathcal T$6 any class of i.i.d.\ laws with $\mathcal T$7 and uniform integrability,

$\mathcal T$8

satisfies $\mathcal T$9. There exists an e-process which diverges on every violating sequence, unifying pathwise SLLNs across $A\subseteq\Omega$0 (Ruf et al., 2022).

• Sequential Testing & Anytime-Valid Inference: Events with $A\subseteq\Omega$1 admit e-processes $A\subseteq\Omega$2 yielding level-$A\subseteq\Omega$3 sequential tests, $A\subseteq\Omega$4, robust under composite nulls.
• Azuma–Hoeffding Bound: For bounded increments, composite Ville recovers pathwise Azuma–Hoeffding inequalities; e.g., $A\subseteq\Omega$5.
• Online Learning: Sequential composite Ville structures supermartingales for regret bounds, recovering algorithms like exponential weights and mirror descent.
• Extensions: Framework supports composite versions of other limit laws (LIL, ergodic theorems), and tighter confidence sequences under minimal moment assumptions.

6. Technical Innovations and Inequalities

The composite Ville framework justifies e-processes as versatile generalizations of martingales, crucial in scenarios where no single process behaves as a martingale under all $A\subseteq\Omega$6. Notable technical results include a new $A\subseteq\Omega$7–type line-crossing inequality for random walks:

$A\subseteq\Omega$8

where $A\subseteq\Omega$9 and only finite first moment is required. This sharpens the pathwise control over deviation probabilities in heavy-tailed, nonparametric regimes (Ruf et al., 2022).

7. Significance and Research Directions

Composite Ville’s theorem not only bridges measure-theoretic and game-theoretic probability, but constitutes the backbone for a minimax duality characterized in game-theoretic frameworks (Frongillo, 24 Dec 2025), advancing robust inference, law-of-large-numbers type results, and sequential testing in settings with uncertainty or adversarial data generation. This suggests further composite generalizations for other probabilistic limit laws and advances the use of e-processes in robust meta-analysis, exchangeability testing, and online learning methodologies. A plausible implication is that future research will refine minimax duality and extend composite Ville’s theorem to infinite-horizon and high-dimensional settings, leveraging e-processes and outer measures for nonparametric testing and confidence sequences.