{
  "id": "2009.02777",
  "version": "v1",
  "published": "2020-09-06T17:04:02.000Z",
  "updated": "2020-09-06T17:04:02.000Z",
  "title": "A note on powers of the characteristic function",
  "authors": [
    "Saulius Norvidas"
  ],
  "comment": "6 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $CH(R)$ denote the family of characteristic functions of probability measures (distributions) on the real line $R$. We study the following question: given an integer $n>1$, do there exist two different $f, g\\in CH(R)$ such that $ f^n\\equiv g^n$? For positive even $n$, well-known examples answer this question in the affirmative. It turns out that the same is true also for any odd $n>1$. For $f\\in CH(R)$ and integer $n>1$, set $C_n(f)=\\{g\\in CH(R): g^n\\equiv f^n\\}$. In this paper, we give an estimate for cardinality (or cardinal number) of $C_n(f)$. In addition, we describe such $f$ for which our estimate is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-06T17:04:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A38",
      "42A82",
      "60E10"
    ],
    "keywords": [
      "characteristic function",
      "well-known examples answer",
      "probability measures",
      "cardinal number",
      "real line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}