On the Monotonicity of Discrete Entropy for Log-Concave Random Vectors on $$\mathbb {Z}^d$$
Abstract
Abstract We prove the following type of discrete entropy monotonicity for sums of isotropic, log-concave, independent and identically distributed random vectors $$X_1,\dots ,X_{n+1}$$ X 1 , ⋯ , X n + 1 on $$\mathbb {Z}^d$$ Z d : $$ H(X_1+\cdots +X_{n+1}) \ge H(X_1+\cdots +X_{n}) + \frac{d}{2}\log {\Bigl (\frac{n+1}{n}\Bigr )} +o(1), $$ H ( X 1 + ⋯ + X n + 1 ) ≥ H ( X 1 + ⋯ + X n ) + d 2 log ( n + 1 n ) + o ( 1 ) , where o (1) vanishes as $$H(X_1) \rightarrow \infty $$ H ( X 1 ) → ∞ . Moreover, for the o (1)-term, we obtain a rate of convergence $$ O\Bigl ({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr )$$ O ( H ( X 1 ) e - 1 d H ( X 1 ) ) , where the implied constants depend on d and n . This generalizes to $$\mathbb {Z}^d$$ 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 $$H(X_1+\cdots +X_{n})$$
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Point processes and geometric inequalities
Mathematical Approximation and Integration
Publication details
- Journal / source
- Discrete & Computational Geometry
- Publisher
- Springer Science and Business Media LLC
- Volume
- 76
- Issue
- 2
- Pages
- 1111–1149
- Publication date
- 26 May 2026
- Language
- EN