The entropy power inequality without assumptions: Equality and stability

We prove that the entropy power inequality (EPI) holds with equality if and only if the variables are Gaussian, with out making any assumptions beyond the existence of entropies. In particular, this shows that the Stam-Blachman argument gives the EPI without requiring any conditions at all. The key step is the justification of de Bruijn’s identity without moment assumptions. We further obtain a qualitative stability result for the EPI, in terms of weak convergence, under very mild moment conditions: Whenever the EPI is close to an equality, the random variables involved are necessarily close to being Gaussian.