Discrete Generalised Entropy Power Inequalities for Log-Concave Random Variables

We prove a discrete analogue of the generalised entropy power inequality for log-concave random variables on the integers. As a special case, we show that a conjecture of Tao (2010) holds true for log-concave random variables on the integers:\begin{equation*}H\left( {{X_1} + \cdot s + {X_{n + 1}}} \right) \geq H\left( {{X_1} + \cdot s + {X_n}} \right) + \frac{1}{2}\log \left( {\frac{{n + 1}}{n}} \right) - o(1),\end{equation*}where the o(1)-term vanishes as H(X
1
) → ∞. Explicit, finite bounds for the error term are provided, which are exponential in H(X
1
).