{
  "id": "2401.12605",
  "version": "v1",
  "published": "2024-01-23T10:04:50.000Z",
  "updated": "2024-01-23T10:04:50.000Z",
  "title": "The asymptotic behavior of fraudulent algorithms",
  "authors": [
    "Michel Benaïm",
    "Laurent Miclo"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $U$ be a Morse function on a compact connected $m$-dimensional Riemannian manifold, $m \\geq 2,$ satisfying $\\min U=0$ and let $\\mathcal{U} = \\{x \\in M \\: : U(x) = 0\\}$ be the set of global minimizers. Consider the stochastic algorithm $X^{(\\beta)}:=(X^{(\\beta)}(t))_{t\\geq 0}$ defined on $N = M \\setminus \\mathcal{U},$ whose generator is$U \\Delta \\cdot-\\beta\\langle \\nabla U,\\nabla \\cdot\\rangle$, where $\\beta\\in\\RR$ is a real parameter.We show that for $\\beta>\\frac{m}{2}-1,$ $X^{(\\beta)}(t)$ converges a.s.\\ as $t \\rightarrow \\infty$, toward a point $p \\in \\mathcal{U}$ and that each $p \\in \\mathcal{U}$ has a positive probability to be selected. On the other hand, for $\\beta < \\frac{m}{2}-1,$ the law of $(X^{(\\beta)}(t))$ converges in total variation (at an exponential rate) toward the probability measure $\\pi_{\\beta}$ having density proportional to $U(x)^{-1-\\beta}$ with respect to the Riemannian measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-23T10:04:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fraudulent algorithms",
      "asymptotic behavior",
      "dimensional riemannian manifold",
      "probability measure",
      "exponential rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}