Stability of the Shannon–Stam inequality via the Föllmer process

Ronen Eldan, Dan Mikulincer

Research output: Contribution to journalArticlepeer-review

Abstract

We prove stability estimates for the Shannon-Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors X, Y. Rd, the deficit in the Shannon-Stam inequality is bounded from below by the expression C (D(X||G) + D(Y ||G)), where D(. ||G) denotes the relative entropy with respect to the standard Gaussian and the constant C depends only on the covariance structures and the spectral gaps of X and Y. In the case of uniformly log-concave vectors our analysis gives dimension-free bounds. Our proofs are based on a new approach which uses an entropy-minimizing process from stochastic control theory.

Original languageEnglish
Pages (from-to)891-922
Number of pages32
JournalProbability Theory and Related Fields
Volume177
Issue number3-4
Early online date11 Mar 2020
DOIs
StatePublished - Aug 2020

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