Abstract
AbstractIt 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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science