{
  "id": "2409.19845",
  "version": "v3",
  "published": "2024-09-30T01:10:40.000Z",
  "updated": "2024-10-23T11:58:42.000Z",
  "title": "Sign changes of the partial sums of a random multiplicative function III: Average",
  "authors": [
    "Marco Aymone"
  ],
  "comment": "9 pages, v3: new results and references added. Connection with a Theorem of Erdos and Hunt has been made",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "Let $V(x)$ be the number of sign changes of the partial sums up to $x$, say $M_f(x)$, of a Rademacher random multiplicative function $f$. We prove that the averaged value of $V(x)$ is at least $\\gg (\\log x)(\\log\\log x)^{-1/2-\\epsilon}$. Our new method applies for the counting of sign changes of the partial sums of a system of orthogonal random variables having variance $1$ under additional hypothesis on the moments of these partial sums. In particular, we recover and extend to larger classes of dependencies an old result of Erd\\H{o}s and Hunt on sign changes of partial sums of i.i.d. random variables. In the arithmetic case, the main input in our method is the ``\\textit{linearity}'' phase in $1\\leq q\\leq 1.9$ of the quantity $\\log \\EE |M_f(x)|^q$, provided by the Harper's \\textit{better than squareroot cancellation} phenomenon for small moments of $M_f(x)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-10-23T11:58:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partial sums",
      "sign changes",
      "rademacher random multiplicative function",
      "orthogonal random variables",
      "arithmetic case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}