Stability of the Shannon-Stam inequality via the Follmer process

• 成果类型: Article
• 署名作者: Eldan, Ronen; Mikulincer, Dan
• 署名单位: Weizmann Institute of Science
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00967-w
• 发表日期: 2020
• 页码: 891-922
• 关键词: entropy jumps

摘要

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.