Extended Ville’s Inequality

• Extended Ville’s Inequality is a generalization of the classical inequality that uses time-varying monotonic barriers to control supermartingale behavior.
• The framework employs 'floor-hugger' martingale constructions to achieve tight bounds, even extending to nonintegrable processes via truncation methods.
• It underpins practical applications such as anytime-valid sequential inference, finite-sample laws, and multiple testing in modern statistical settings.

The extended Ville’s inequality encompasses a spectrum of maximal inequalities for supermartingales, generalizing the original result to richer settings including monotonic lower bounds, time-varying upper thresholds, and nonintegrable processes. These generalizations are fundamental in the analysis of anytime-valid inference, sequential hypothesis testing, and the structure of e-processes within modern sequential statistics.

1. Classical Ville's Inequality: Background and Scope

Ville’s inequality provides a foundational probabilistic tail bound for nonnegative supermartingales. Given a filtered probability space and a nonnegative adapted supermartingale $\{M_n\}_{n\ge0}$, the classical result asserts that for any constant $C>0$,

$$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$

This bound is central in avoiding a union bound when establishing type I error control under optional stopping and is key in constructing confidence sequences and e-values (Koolen et al., 22 Feb 2025, Wang et al., 2023).

2. Generalization to Monotonic Barriers and Thresholds

The generalisation of Ville’s inequality replaces the constant lower and upper bounds with monotonic curves $a(n)$ and $b(n)$, respectively. Specifically, $a(n)$ is a nonincreasing process representing a time-varying floor, and $b(n)$ a nondecreasing process representing a moving upper threshold. Under this framework, the key inequality is:

$$\begin{aligned} &\text{Let } a: \mathbb{N} \to \mathbb{R} \text{ nonincreasing},~ b: \mathbb{N} \to \mathbb{R} \text{ nondecreasing},\ & a(0) < b(0),~ M_0 \in [a(0), b(0)]~\text{a.s.},~ \mathbb{E}[M_0] \in [a(0), b(0)],\ & (M_n)_{n\ge0}~\text{is adapted with}~\mathbb{E}[M_{n+1}|\mathcal{F}_n]\le M_n,~M_n\ge a(n)~\forall n, \end{aligned}$$

then defining

$$s(n) = \prod_{k=n}^\infty \left(1-\frac{a(k)-a(k+1)}{b(k+1)-a(k+1)}\right),$$

the general bound is

$$\Pr\Bigl(\exists n \ge 0 : M_n \ge b(n)\Bigr) \le 1-\frac{b(0)-\mathbb{E}[M_0]}{b(0)-a(0)}\prod_{k=0}^\infty\frac{b(k+1)-a(k)}{b(k+1)-a(k+1)}.$$

When $C>0$0, $C>0$1, the classical bound is retrieved (Koolen et al., 22 Feb 2025).

3. Tightness and “Floor-Hugger” Martingale Construction

The sharpness of the extended inequality is established through the construction of a martingale that achieves bound equality, often termed the “floor-hugger” martingale. This process starts at $C>0$2, remains at the floor $C>0$3 as long as possible, and at each time $C>0$4, with probability

$C>0$5

jumps to the current threshold $C>0$6. Otherwise, it continues hugging the floor. This construction produces the exact product-form hitting probability prescribed by the general bound, demonstrating optimality (Koolen et al., 22 Feb 2025).

4. Extensions to Nonintegrable Supermartingales

The theory of extended Ville’s inequality further loosens integrability requirements by considering extended nonnegative supermartingales (ENSMs) taking values in $C>0$7. For such $C>0$8, the following bound holds for all $C>0$9 and all $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$0: $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$1 This formulation maintains validity even when $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$2, by replacing expectations with their truncated counterpart. The proof uses truncation and extended conditional expectation theory, preserving martingale properties across arbitrary stopping times (Wang et al., 2023).

Additionally, improper or $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$3-finite mixtures of ENSMs—often arising from “flat” priors—preserve the supermartingale structure and validity of the inequality. This is instrumental for modern sequential testing and construction of confidence sequences under nonparametric or composite nulls.

5. Applications: Sequential Inference and Concentration

Several applications are enabled by the extended inequality:

• Anytime-valid sequential inference: Construction of confidence sequences and e-values that retain error guarantees under arbitrary stopping, avoiding union bounds. For example, combining with improper mixture arguments yields “flat prior” martingales valid for confidence sequences in nonparametric mean estimation (Wang et al., 2023).
• Finite-time Law of the Iterated Logarithm: By selecting appropriate $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$4 and $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$5, the generalised inequality produces finite-sample versions of the LIL with explicit constants. For i.i.d. sub-Gaussian variables,

$$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$6

holds with probability at least $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$7 for every $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$8, with $$\Pr\bigl(\exists n \ge 0 : M_n \ge C\bigr) \le \frac{\mathbb{E}[M_0]}{C}.$$9, recovering the classic iterated-log asymptotics up to vanishing $a(n)$0 terms (Koolen et al., 22 Feb 2025).

• Multiple testing and subsidized paths: The nonconstant lower barrier $a(n)$1 can be interpreted as a subsidy, modeling scenarios where repeated or parallel testing is allowed, with the bound naturally adjusting to account for the subsidy effect on hitting probabilities (Koolen et al., 22 Feb 2025).
• Game-theoretic probability and e-processes: The result encompasses e-processes and improper mixtures in defensive forecasting. Extended inequalities are crucial when standard supermartingale mixtures become improper or dip below zero in a controlled fashion (Koolen et al., 22 Feb 2025, Wang et al., 2023).

6. Technical Summary and Fundamental Implications

The following table summarizes key distinctions and features of the various Ville-type inequalities:

Inequality Type	Integrability Required	Allowable Barriers	Typical Application
Classical Ville	Yes	$a(n)$2, $a(n)$3	Optional stopping, e-values
Monotonic-barrier Ville (Koolen et al., 22 Feb 2025)	Yes	$a(n)$4 nonincreasing, $a(n)$5 nondecreasing	Finite-time LIL, subsidized testing
Nonintegrable extended Ville (Wang et al., 2023)	No	$a(n)$6, $a(n)$7	Improper mixtures, nonparametrics

This framework unifies anytime-valid methodology across integrable and improper (non-integrable) supermartingale regimes and enables the design of valid and sharp controls for pathwise exceeding events under complex, time-varying constraints. The interplay with mixture methods allows for rigorous nonparametric inference even with improper priors.

7. Connections and Ongoing Directions

The advances in extended Ville-type inequalities resolve several foundational issues in the construction of optional-stopping–robust methods for sequential analysis. The incorporation of monotonic boundaries and the allowance for nonintegrable processes extends the reach to settings involving nonparametric testing, e-processes for composite nulls, and robust uncertainty quantification via confidence sequences.

A plausible implication is the potential to further generalize optional stopping inequalities to continuous time, more general filtrations, and settings where supermartingale lower bounds are stochastic rather than deterministic. These developments are tightly coupled with advances in e-process theory, game-theoretic probability, and sequential multiple testing procedures (Koolen et al., 22 Feb 2025, Wang et al., 2023).