Approximate discrete entropy monotonicity for log-concave sums

Abstract

Abstract It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$ , if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then \begin{equation*} H(X_1+\cdots +X_{n+1}) \geq H(X_1+\cdots +X_{n}) + \frac {1}{2}\log {\Bigl (\frac {n+1}{n}\Bigr )} - o(1) \end{equation*} as $H(X_1) \to \infty$ , where $H(X_1)$ denotes the (discrete) Shannon entropy. The problem is reduced to the continuous setting by showing that if $U_1,\ldots,U_n$ are independent continuous uniforms on $(0,1)$ , then \begin{equation*} h(X_1+\cdots +X_n + U_1+\cdots +U_n) = H(X_1+\cdots +X_n) + o(1), \end{equation*} as $H(X_1) \to \infty$ , where $h$ stands for the differential entropy. Explicit bounds for the $o(1)$ -terms are provided.