Entropy power inequality

For a random vector ${\displaystyle X:\Omega \to \mathbb {R} ^{n}}$ with probability density function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$, the differential entropy of ${\displaystyle X}$, denoted ${\displaystyle h(X)}$, is defined to be

${\displaystyle h(X)=-\int _{\mathbb {R} ^{n}}f(x)\log f(x)\,dx}$

and the entropy power of ${\displaystyle X}$, denoted ${\displaystyle N(X)}$, is defined to be

${\displaystyle N(X)={\frac {1}{2\pi e}}e^{{\frac {2}{n}}h(X)}.}$

In particular, ${\displaystyle N(X)=|K|^{1/n}}$ when ${\displaystyle X}$ is normally distributed with covariance matrix ${\displaystyle K}$.

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be independent random variables with probability density functions in the ${\displaystyle L^{p}}$ space ${\displaystyle L^{p}(\mathbb {R} ^{n})}$ for some ${\displaystyle p>1}$. Then

${\displaystyle N(X+Y)\geq N(X)+N(Y).}$

Moreover, equality holds if and only if ${\displaystyle X}$ and ${\displaystyle Y}$ are multivariate normal random variables with proportional covariance matrices.