{
  "id": "2401.15462",
  "version": "v1",
  "published": "2024-01-27T17:05:00.000Z",
  "updated": "2024-01-27T17:05:00.000Z",
  "title": "On the monotonicity of discrete entropy for log-concave random vectors on $\\mathbb{Z}^d$",
  "authors": [
    "Matthieu Fradelizi",
    "Lampros Gavalakis",
    "Martin Rapaport"
  ],
  "comment": "24 pages, no figures",
  "categories": [
    "math.PR",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "We prove the following type of discrete entropy monotonicity for isotropic log-concave sums of independent 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. However, 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, 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-27T17:05:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "94A17",
      "52C07",
      "39B62"
    ],
    "keywords": [
      "log-concave random vectors",
      "identically distributed random vectors",
      "log-concave function",
      "identically distributed uniform random vectors",
      "independent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}