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 language | English |
---|---|
Pages (from-to) | 891-922 |
Number of pages | 32 |
Journal | Probability Theory and Related Fields |
Volume | 177 |
Issue number | 3-4 |
Early online date | 11 Mar 2020 |
DOIs | |
State | Published - Aug 2020 |