A Bayesian Proof of the Bernoulli Theorem

2026-08-11

AI Summary

This work revisits the Bernoulli law of large numbers from an information-theoretic perspective and characterizes the fundamental limit of the supremum of Bernoulli processes. By introducing Bayesian estimation and leveraging the intrinsic constraints of Cauchy additive channels, the study establishes, for the first time, an information-theoretic lower bound on the supremum of Bernoulli processes and proposes a novel information-theoretic functional analogous to the Fernique functional. Combining coupling techniques for stochastic processes with tools from functional analysis, this approach not only yields a new proof of Bernoulli’s theorem but also extends the phenomenon of distributional reinforcement—previously known in Gaussian processes—to the Bernoulli setting, thereby constructing a comprehensive information-theoretic framework for characterizing the expected maximum over arbitrary index sets.

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.

Problem

• Bernoulli process
• supremum
• Bayesian estimation
• majorizing measure
• coupling

Innovation

• Bernoulli process
• information-theoretic functional
• Bayesian estimation
• supremum characterization
• Cauchy additive channel