A Bayesian Proof of the Bernoulli Theorem

Abstract

We give a new proof of the Bernoulli theorem, conjectured by Talagrand and proved in the seminal work of Bednorz and Latała. Our approach is based on information-theoretic ideas: lower bounds on the supremum of a Bernoulli process are translated to the fundamental limits of Bayesian estimation in a Cauchy additive channel. This leads to a new information-theoretic functional that characterizes Bernoulli-process suprema and plays a role analogous to Fernique's majorizing-measure functional for Gaussian processes. The same viewpoint yields a distributional strengthening: for any prescribed law of the index, we characterize the largest expected value attainable over all couplings of that index with the Bernoulli process. This extends to Bernoulli processes a phenomenon previously understood for Gaussian processes through the work of Fernique and Talagrand.

Disclosure

“ctional. Section 5 converts this functional into the ℓ1 -plus-Gaussian decomposition, and Section 6 proves the prescribed-law distributional result. The Cauchy information–estimation inequality is proved in Appendix A. Statement on AI use. ChatGPT Pro (versions 5.5 and 5.6) was used to generate an initial proof sketch for the information-estimation inequality in Appendix A, specifically in response to the authors’ query regarding whether the inequality holds for the Cauchy additive”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:09:48.132686Z

Structural counts