Information on trajectories: martingales and random times

2026-08-20

AI Summary

研究通过路径空间信息流，使用变分恒等式方法解决非负鞅在随机时间点的精确控制问题，并分析了多种几何形态下的舍弃松弛。

Abstract

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 $L^p$ 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.

Problem

• martingales
• random times
• information flow
• concentration inequalities
• variational identities

Innovation

• martingales
• variational identities
• concentration inequalities
• Bregman divergences
• safe testing