Ville’s Inequality: Extensions & Applications

• Ville’s inequality is a maximal inequality for nonnegative supermartingales, extending Markov’s inequality to control the full trajectory of stochastic processes.
• Recent advancements remove classical integrability constraints and extend the inequality to improper mixtures and moving boundaries, enhancing sequential statistical methods.
• The framework supports anytime-valid inference, composite testing, and nonparametric confidence sequences, impacting e-processes and finite-time LIL results.

Ville’s inequality is a fundamental maximal inequality in probability, governing the likelihood that a nonnegative supermartingale process ever exceeds a threshold. It generalizes Markov’s inequality to the full trajectory of stochastic processes, underpinning a wide class of anytime-valid methods in sequential analysis, online testing, and nonparametric inference. Recent advances have established sharp extensions of Ville’s inequality to non-integrable supermartingales, improper mixtures, and monotonic boundaries, yielding new statistical tools and generalizations in sequential statistics, e-processes, and nonparametric confidence sequences (Wang et al., 2023, Koolen et al., 22 Feb 2025).

1. Classical Formulation and Connection to Markov’s Inequality

The classical Ville’s inequality states that for a nonnegative supermartingale $(M_n)_{n\geq 0}$ adapted to a filtration $(\mathcal{F}_n)$ on $(\Omega, \mathcal{F}, \mathbb{P})$, if $M_0 \geq 0$ a.s. and $\mathbb{E}[M_n] < \infty$, then for any $C > 0$,

$$\Pr\left(\exists n \geq 0: M_n \geq C\right) \leq \frac{\mathbb{E}[M_0]}{C}.$$

Markov’s inequality is the special case $n=0$: $\Pr(M_0 \geq C) \leq \frac{\mathbb{E}[M_0]}{C}.$ Ville’s inequality thus extends the reach of Markov’s bound from a single time point to the supremum over all times, without incurring a union-bound penalty (Koolen et al., 22 Feb 2025).

2. Supermartingale Structures and Integrability

A process $(M_n)$ is a nonnegative supermartingale (NSM) if:

• $(\mathcal{F}_n)$0 is $(\mathcal{F}_n)$1-measurable;
• $(\mathcal{F}_n)$2 a.s.;
• $(\mathcal{F}_n)$3;
• $(\mathcal{F}_n)$4 a.s. for all $(\mathcal{F}_n)$5.

In classical treatments, integrability ($(\mathcal{F}_n)$6) is required for all $(\mathcal{F}_n)$7. Recent work has demonstrated that this assumption can be removed by defining extended nonnegative supermartingales (ENSMs), where $(\mathcal{F}_n)$8 may be infinite and expectations are defined via monotone limits: $(\mathcal{F}_n)$9 with $(\Omega, \mathcal{F}, \mathbb{P})$0 (Wang et al., 2023).

3. Extensions to Nonintegrable and Monotonic Boundaries

3.1 Extended Inequality for ENSMs

For an ENSM $(\Omega, \mathcal{F}, \mathbb{P})$1 and any $(\Omega, \mathcal{F}, \mathbb{P})$2, the extended Ville’s inequality is

$(\Omega, \mathcal{F}, \mathbb{P})$3

which remains valid even when $(\Omega, \mathcal{F}, \mathbb{P})$4. This is achieved by considering the process $(\Omega, \mathcal{F}, \mathbb{P})$5 and applying the classical result (Wang et al., 2023).

3.2 Generalization to Moving Boundaries

If $(\Omega, \mathcal{F}, \mathbb{P})$6 is a nonnegative supermartingale, and $(\Omega, \mathcal{F}, \mathbb{P})$7 (decreasing) and $(\Omega, \mathcal{F}, \mathbb{P})$8 (increasing) are deterministic curves satisfying $(\Omega, \mathcal{F}, \mathbb{P})$9, the generalized Ville-type inequality for the event

$M_0 \geq 0$0

is

$M_0 \geq 0$1

For finite stopping time $M_0 \geq 0$2,

$M_0 \geq 0$3

provided that $M_0 \geq 0$4 is strictly positive and differentiable (Koolen et al., 22 Feb 2025).

4. The Extended Method of Mixtures

Given a $M_0 \geq 0$5-finite family of ENSMs $M_0 \geq 0$6 and a $M_0 \geq 0$7-finite measure $M_0 \geq 0$8 over $M_0 \geq 0$9, the mixed process defined by

$\mathbb{E}[M_n] < \infty$0

is itself an ENSM, as shown via Tonelli's theorem. This extension supports improper priors and improper mixtures, and is especially significant in nonparametric Bayesian analysis and the construction of e-processes (Wang et al., 2023).

5. Tightness and Extremal Supermartingale Constructions

The generalization to monotonic boundaries is proved to be tight by explicit construction of a "floor-hugger" martingale. In discrete time, the process starts at the floor $\mathbb{E}[M_n] < \infty$1 and moves by either dropping to the next floor $\mathbb{E}[M_n] < \infty$2 or jumping to the threshold $\mathbb{E}[M_n] < \infty$3, with probability chosen so that

$\mathbb{E}[M_n] < \infty$4

This process never exceeds the threshold except possibly at a single jump, and the probability of stay-below equals the upper bound, confirming sharpness (Koolen et al., 22 Feb 2025).

6. Statistical Applications

6.1 Anytime-Valid Confidence Sequences

• Classical (proper mixture): For 1-subGaussian $\mathbb{E}[M_n] < \infty$5, mixing the likelihood ratio supermartingale over $\mathbb{E}[M_n] < \infty$6 yields valid confidence sequences via the classical Ville bound.
• Improper mixture: Integrating the likelihood ratio against an improper uniform prior yields a process $\mathbb{E}[M_n] < \infty$7 that is an ENSM but not integrable. The extended Ville bound produces valid nonparametric confidence sequences even in this improper setting: $\mathbb{E}[M_n] < \infty$8 (Wang et al., 2023).

6.2 Composite Testing and Extended e-Processes

For composite nulls, extended e-processes $\mathbb{E}[M_n] < \infty$9 satisfy $C > 0$0 under each $C > 0$1 with $C > 0$2 an ENSM. The mixture lemma guarantees that mixtures of extended e-processes remain in the class, accommodating improper priors and yielding tight, nonparametric anytime-valid bounds for composite hypotheses (Wang et al., 2023).

6.3 Nonparametric Finite-Time Law of the Iterated Logarithm

Applying the generalised Ville bound, process construction with appropriately chosen moving floors and thresholds yields finite-time versions of the law of the iterated logarithm (LIL) for sums of subGaussian variables: $C > 0$3 holding uniformly for all $C > 0$4 with high probability and circumventing the need for delicate slicing techniques or mixture over $C > 0$5 (Koolen et al., 22 Feb 2025).

7. Impact, Modularity, and Implications

Modern formulations of Ville’s inequality unify the treatment of integrable and nonintegrable supermartingales, enable the use of improper mixtures and moving boundaries, and lead to sharpened statistical tools such as anytime-valid inference, composite sequential tests, and nonparametric confidence sequences. Significantly, the modularity of these approaches streamlines proof techniques and assignment of validities without union bounds, and enables the immediate derivation of nonparametric results and tightness guarantees. This framework underlies the theory of e-processes and sharp law-of-iterated-logarithm-type results, which are now foundational in the asymptotic and finite-sample regimes of sequential analysis (Wang et al., 2023, Koolen et al., 22 Feb 2025).