# Information on trajectories: martingales and random times

arXiv: https://arxiv.org/abs/2608.20337 · HTML: https://mlsanity.com/paper/2608.20337/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 54 references that are in the map.
- First posted on arXiv 20 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [On the near-optimality of betting confidence sets for bounded means](https://mlsanity.com/paper/2310.01547/) (2023)
- [Information from coincidences](https://mlsanity.com/paper/2606.25042/) (2026)
- [p-value peeking and estimating extrema](https://mlsanity.com/paper/2011.01343/) (2020)

## Abstract (arXiv)

Accounting for information flow on the path space of trajectories of a nonnegative martingale yields exact variational identities for it, even at arbitrary random times. This recovers the widely used classical concentration inequalities, from Ville to PAC-Bayes, and measures what each one discards. The tail a bound controls is itself a relative entropy, resolved by the chain rule into per-step conditional divergences. The discarded slack has an exact form in each of three geometries: a Gibbs tilt for the Azuma-Hoeffding and PAC-Bayes bounds, the crossing itself for Ville's and for pooled tests, and a dominating certificate for the Lp maximal bound. That certificate's optional-stopping deficit resolves per step into Bregman divergences of the running maximum. On a path-time space, the same identity gains one factor that prices anticipation: an arbitrary random time carries an e-process ``peeking penalty.'' The partition function can be read as a coalescent--a prefix-sharing probability of independent copies--and geometric mixtures of test martingales gain a pooling benefit for multi-model safe testing.