{
  "id": "2204.10681",
  "version": "v1",
  "published": "2022-04-22T12:53:29.000Z",
  "updated": "2022-04-22T12:53:29.000Z",
  "title": "A Weak Law of Large Numbers for Dependent Random Variables",
  "authors": [
    "Ioannis Karatzas",
    "Walter Schachermayer"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Every sequence $f_1, f_2, \\cdots \\, $ of random variables with $ \\, \\lim_{M \\to \\infty} \\big( M \\sup_{k \\in \\mathbb{N}} \\mathbb{P} ( |f_k| > M ) \\big)=0\\,$ contains a subsequence $ f_{k_1}, f_{k_2} , \\cdots \\,$ that satisfies, along with all its subsequences, the weak law of large numbers: $ \\, \\lim_{N \\to \\infty} \\big( (1/N) \\sum_{n=1}^N f_{k_n} - D_N \\big) =0\\,,$ in probability. Here $\\, D_N\\, $ is a \"corrector\" random variable with values in $[-N,N]$, for each $N \\in \\mathbb{N} $; these correctors are all equal to zero if, in addition, $\\, \\liminf_{k \\to \\infty} \\mathbb{E} \\big( f_k^2 \\, \\mathbf{ 1}_{ \\{ |f_k| \\le M \\} } \\big) =0\\,$ holds for every $M \\in (0, \\infty)\\,.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-22T12:53:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60A10",
      "60F15"
    ],
    "keywords": [
      "dependent random variables",
      "large numbers",
      "weak law",
      "subsequence",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}