  The entropy power inequality (EPI) provides lower bounds on the differential
entropy of the sum of two independent real-valued random variables in terms of
the individual entropies. Versions of the EPI for discrete random variables
have been obtained for special families of distributions with the differential
entropy replaced by the discrete entropy, but no universal inequality is known
(beyond trivial ones). More recently, the sumset theory for the entropy
function provides a sharp inequality $H(X+X')-H(X)\geq 1/2 -o(1)$ when $X,X'$
are i.i.d. with high entropy. This paper provides the inequality $H(X+X')-H(X)
\geq g(H(X))$, where $X,X'$ are arbitrary i.i.d. integer-valued random
variables and where $g$ is a universal strictly positive function on $\mR_+$
satisfying $g(0)=0$. Extensions to non identically distributed random variables
and to conditional entropies are also obtained.
