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 $\mu^*(A)=0$.

• $\mathcal P$–e-Process: A nonnegative, $\mathcal F_t$–adapted process $E=(E_t)_{t\ge0}$ with $E_0=1$ such that for every stopping time $\tau\in\mathcal T$,

$\sup_{P\in\mathcal P} E_P[E_\tau] \leq 1.$

For every $P\in\mathcal P$, there exists a nonnegative $P$–martingale $M^P$ with $E_t \leq M^P_t$ almost surely for all $t$.

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:

$$\boxed{ \mu^*(A)=0 \quad\Longleftrightarrow\quad \exists\ {\rm a\ } \mathcal P{\text{--e--process}~}E {\text{~with~}} E_t \to \infty \text{~on~}A }$$

More generally, for $a\in(0,1]$,

$$\mu^*(A)\le a \quad\Longleftrightarrow\quad \exists\ \mathcal P{\text{--e--process}~}E {\text{~with~}} \sup_t E_t \ge 1/a \text{~on~}A.$$

Outline of proof:

• If $\mu^*(A) = 0$: For each $n$, exist $\tau_n$ so that $A \subseteq \{\tau_n < \infty\}$ and $\sup_P P(\tau_n < \infty) \le 2^{-n}$. Construct $$E_t = \sum_{n=1}^\infty \mathbf{1}_{\{\tau_n \le t\}}$$; this diverges on $A$ and satisfies the e-process expectation bound.
• Only if $E_t \to \infty$ on $A$: For $a > 0$, let $\tau = \inf\{ t: E_t \ge 1/a \}$. On $A$, $\tau < \infty$. Using the e-process property, $\sup_P P(\tau<\infty) \le a$. As $a \to 0$, this ensures $\mu^*(A) = 0$ (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)	$$\inf_{Z\in Z} \sup_{\omega\in\Omega} (X(\omega)-Z(\omega))$$	betting replication
Measure-theoretic (World first)	$\sup_{P} \inf_{Z\in Z} E_P[X-Z]$	probabilistic
Composite (worst-case P)	$\sup_{P\in \Delta_0} E_P[X]$	robust consistency

Minimax duality for sequential gambling states:

$$\sup_{P \in \Delta_0} E_P[X] = \inf_{\psi} \sup_{\omega} (X(\omega) - Z_T^{\psi}(\omega))$$

where $Z_T^\psi$ is the cumulative payoff of a strategy $\psi$. 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 $\mathcal P = \{P\}$, $\mu^*(A) = P(A)$ and $\mathcal P$–e–processes are precisely $P$–martingales. The theorem reduces to: $P(A)=0$ iff exists nonnegative martingale $M_t \to \infty$ on $A$.
• For general $\mathcal P$, inverse capital measure replaces probability, and e-processes replace martingales. Stopping times become composite tests with uniform type-I error control across $\mathcal P$ (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 $\mathcal P$ any class of i.i.d.\ laws with $\mathbb E_P|X_1|<\infty$ and uniform integrability,

$$A_{\rm div} = \left\{ \lim_{t\to\infty} \frac{1}{t}\sum_{s=1}^t X_s \text{ does not exist} \right\}$$

satisfies $\mu^*(A_{\rm div})=0$. There exists an e-process which diverges on every violating sequence, unifying pathwise SLLNs across $\mathcal P$ (Ruf et al., 2022).

• Sequential Testing & Anytime-Valid Inference: Events with $\mu^*(A)=0$ admit e-processes $E_t$ yielding level-$\alpha$ sequential tests, $\{\sup_{s\le t} E_s \ge 1/\alpha\}$, robust under composite nulls.
• Azuma–Hoeffding Bound: For bounded increments, composite Ville recovers pathwise Azuma–Hoeffding inequalities; e.g., $$\sup_{P\in\Delta_0} P(\sum Y_t \ge \epsilon) \le e^{-\epsilon^2/(2T)}$$.
• 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 $\mathcal P$. Notable technical results include a new $\mathbf L^1$–type line-crossing inequality for random walks:

$$\mathbb P\left(\sup_{t\ge1} \frac{|\sum_{s=1}^t X_s|}{\gamma + t} > \epsilon + r(K) \right) \le \frac{8K^2}{\gamma\epsilon^2} + \left( \frac{16}{\epsilon^2} + 2 \right) r(K)$$

where $r(K) = \mathbb E[|X_1|\mathbf{1}_{|X_1|>K}]$ 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.