# The extended Ville's inequality for nonintegrable nonnegative supermartingales

arXiv: https://arxiv.org/abs/2304.01163 · HTML: https://mlsanity.com/paper/2304.01163/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 35 references that are in the map.
- First posted on arXiv 3 Apr 2023.

## Builds on (most cited first, 12 of 12)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [A unified recipe for deriving (time-uniform) PAC-Bayes bounds](https://mlsanity.com/paper/2302.03421/) (2023)
- [Confidence sequences for sampling without replacement](https://mlsanity.com/paper/2006.04347/) (2020)
- [Martingale Methods for Sequential Estimation of Convex Functionals and Divergences](https://mlsanity.com/paper/2103.09267/) (2021)
- [A composite generalization of Ville's martingale theorem using e-processes](https://mlsanity.com/paper/2203.04485/) (2022)
- [Comparing Sequential Forecasters](https://mlsanity.com/paper/2110.00115/) (2021)
- [E-Statistics, Group Invariance and Anytime Valid Testing](https://mlsanity.com/paper/2208.07610/) (2022)

## Built on by (most cited first, 3 of 3)

- [Anytime-valid, Bayes-assisted, Prediction-Powered Inference](https://mlsanity.com/paper/2505.18000/) (2025)
- [Combining Evidence Across Filtrations](https://mlsanity.com/paper/2402.09698/) (2024)
- [Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance](https://mlsanity.com/paper/2310.03722/) (2023)

## Abstract (arXiv)

Following the initial work by Robbins, we rigorously present an extended theory of nonnegative supermartingales, requiring neither integrability nor finiteness. In particular, we derive a key maximal inequality foreshadowed by Robbins, which we call the extended Ville's inequality, that strengthens the classical Ville's inequality (for integrable nonnegative supermartingales), and also applies to our nonintegrable setting. We derive an extension of the method of mixtures, which applies to σ-finite mixtures of our extended nonnegative supermartingales. We present some implications of our theory for sequential statistics, such as the use of improper mixtures (priors) in deriving nonparametric confidence sequences and (extended) e-processes.