{
  "id": "1909.09576",
  "version": "v1",
  "published": "2019-09-20T15:57:37.000Z",
  "updated": "2019-09-20T15:57:37.000Z",
  "title": "On almost sure convergence of random variables with finite chaos decomposition",
  "authors": [
    "Radosław Adamczak"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "Under mild conditions on a family of independent random variables $(X_n)$ we prove that almost sure convergence of a sequence of tetrahedral polynomial chaoses of uniformly bounded degrees in the variables $(X_n)$ implies the almost sure convergence of their homogeneous parts. This generalizes a recent result due to Poly and Zheng obtained under stronger integrability conditions. In particular for i.i.d. sequences we provide a simple necessary and sufficient condition for this property to hold. We also discuss similar phenomena for sums of multiple stochastic integrals with respect to Poisson processes, answering a question by Poly and Zheng.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-20T15:57:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F99",
      "60H05",
      "60B11"
    ],
    "keywords": [
      "sure convergence",
      "finite chaos decomposition",
      "independent random variables",
      "tetrahedral polynomial chaoses",
      "stronger integrability conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}