We prove the following type of discrete entropy monotonicity for sums of isotropic, log-concave, independent and identically distributed random vectors $X_1,\dots,X_{n+1}$ on $\mathbb{Z}^d$: $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{d}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} +o(1), $$ where $o(1)$ vanishes as $H(X_1) \to \infty$. Moreover, for the $o(1)$-term, we obtain a rate of convergence $ O\Bigl({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr)$, where the implied constants depend on $d$ and $n$. This generalizes to $\mathbb{Z}^d$ the one-dimensional result of the second named author (2023). As in dimension one, our strategy is to establish that the discrete entropy $H(X_1+\cdots+X_{n})$ is close to the differential (continuous) entropy $h(X_1+U_1+\cdots+X_{n}+U_{n})$, where $U_1,\dots, U_n$ 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. In fact, we show this result under more general assumptions than log-concavity, which are preserved up to constants under convolution. In order to show that log-concave distributions satisfy our assumptions in dimension $d\ge2$, more involved tools from convex geometry are needed because a suitable position is required. We show that, for a log-concave function on $\mathbb{R}^d$ in isotropic position, its integral, barycenter and covariance matrix are close to their discrete counterparts. Moreover, in the log-concave case, we weaken the isotropicity assumption to what we call almost isotropicity. One of our technical tools is a discrete analogue to the upper bound on the isotropic constant of a log-concave function, which extends to dimensions $d\ge1$ a result of Bobkov, Marsiglietti and Melbourne (2022).

翻译：我们证明了$\mathbb{Z}^d$上各向同性、对数凹、独立同分布随机向量$X_1,\dots,X_{n+1}$的和具有如下形式的离散熵单调性：$$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{d}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} +o(1), $$ 其中$o(1)$项随$H(X_1) \to \infty$而趋于零。此外，对于$o(1)$项，我们得到了收敛速率$ O\Bigl({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr)$，其中隐含常数依赖于$d$和$n$。这将第二作者（2023）的一维结果推广到了$\mathbb{Z}^d$上。与一维情形类似，我们的策略是证明离散熵$H(X_1+\cdots+X_{n})$接近于微分（连续）熵$h(X_1+U_1+\cdots+X_{n}+U_{n})$，其中$U_1,\dots, U_n$是$[0,1]^d$上的独立同分布均匀随机向量，并应用Artstein、Ball、Barthe和Naor（2004）关于微分熵单调性的定理。事实上，我们在比对数凹性更一般的假设下证明了这一结果，这些假设在卷积下能保持到常数倍。为了证明对数凹分布在维度$d\ge2$时满足我们的假设，需要凸几何中更复杂的工具，因为需要合适的定位。我们证明，对于处于各向同性位置上的$\mathbb{R}^d$对数凹函数，其积分、重心和协方差矩阵都与其离散对应项接近。此外，在对数凹情形下，我们将各向同性假设弱化为所谓的几乎各向同性。我们的一个技术工具是对数凹函数各向同性常数上界的离散模拟，这将Bobkov、Marsiglietti和Melbourne（2022）的结果扩展到了$d\ge1$维。