{
  "id": "1401.8012",
  "version": "v1",
  "published": "2014-01-30T22:27:37.000Z",
  "updated": "2014-01-30T22:27:37.000Z",
  "title": "Regular variation of infinite series of processes with random coefficients",
  "authors": [
    "Raluca Balan"
  ],
  "comment": "19 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we consider a series $X(t)=\\sum_{j \\geq 1}\\Psi_j(t) Z_j(t),t \\in [0,1]$ of random processes with sample paths in the space $D=D[0,1]$ of c\\`adl\\`ag functions (i.e. right-continuous functions with left limits) on $[0,1]$. We assume that $(Z_j)_{j \\geq 1}$ are i.i.d. processes with sample paths in $D$ and $(\\Psi_j)_{j \\geq 1}$ are processes with continuous sample paths. Using the notion of regular variation for $D$-valued random elements (introduced in Hult and Lindskog (2005)), we show that $X$ is regularly varying if $Z_1$ is regularly varying, $(\\Psi_j)_{j \\geq 1}$ satisfy some moment conditions, and a certain ``predictability assumption'' holds for the sequence $\\{(Z_j,\\Psi_j)\\}_{j \\geq 1}$. Our result can be viewed as an extension of Theorem 3.1 of Hult and Samorodnitsky (2008) from random vectors in $R^d$ to random elements in $D$. As a preliminary result, we prove a version of Breiman's lemma for $D$-valued random elements, which can be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-30T22:27:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G52",
      "60G17",
      "62M10"
    ],
    "keywords": [
      "regular variation",
      "infinite series",
      "random coefficients",
      "valued random elements",
      "moment conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.8012B"
    }
  }
}