On the monotonicity of discrete entropy for log-concave random vectors on ℤ^d

We prove the following type of discrete entropy monotonicity for isotropic log-concave sums of independent identically distributed random vectors X_1,…,X_n+1 on ℤ^d: H(X_1+⋯+X_n+1) ≥ H(X_1+⋯+X_n) + d/2log(n+1/n) +o(1), where o(1) vanishes as H(X_1) →∞. Moreover, for the o(1)-term we obtain a rate of convergence O(H(X_1)e^-1/dH(X_1)), where the implied constants depend on d and n. This generalizes to ℤ^d the one-dimensional result of the second named author (2023). As in dimension one, our strategy is to establish that the discrete entropy H(X_1+⋯+X_n) is close to the differential (continuous) entropy h(X_1+U_1+⋯+X_n+U_n), where U_1,…, U_n are independent and identically distributed uniform random vectors on [0,1]^d and to apply the theorem of Artstein, Ball, Barthe and Naor (2004) on the monotonicity of differential entropy. However, in dimension d≥2, more involved tools from convex geometry are needed because a suitable position is required. We show that for a log-concave function on ℝ^d in isotropic position, its integral, its barycenter and its covariance matrix are close to their discrete counterparts. One of our technical tools is a discrete analogue to the upper bound on the isotropic constant of a log-concave function, which generalises a result of Bobkov, Marsiglietti and Melbourne (2022) and may be of independent interest.