Information on Trajectories: Martingales and Random Times

Summary

This paper by Akshay Balsubramani (arXiv:2608.20337) studies information flow on the path space of nonnegative martingale trajectories, deriving exact variational identities that hold even at arbitrary random times. The work recovers and unifies classical concentration inequalities from Ville to PAC-Bayes, and quantifies the information discarded by each. The quantity controlled by tail bounds is itself a relative entropy, which decomposes via the chain rule into per-step conditional divergences. The discarded slack takes exact forms under three geometries: Azuma-Hoeffding and PAC-Bayes bounds correspond to Gibbs tilting, Ville bounds and anytime-valid tests correspond to the crossing itself, and L^p maximal bounds correspond to a dominating certificate. The anytime-stopping deficit of this certificate decomposes step by step into Bregman divergences of the running maximum. On path-time space, the same identity gains a factor pricing expectations: any random time carries an e-process peeking penalty.