{
  "id": "1006.1073",
  "version": "v2",
  "published": "2010-06-05T21:52:05.000Z",
  "updated": "2015-05-19T20:23:06.000Z",
  "title": "On the functional limits for partial sums under stable law",
  "authors": [
    "Khurelbaatar Gonchigdanzan",
    "Kamil Marcin Kosiński"
  ],
  "journal": "Statistics and Probability Letters 79 (2009) 1818--1822",
  "doi": "10.1016/j.spl.2009.05.004",
  "categories": [
    "math.PR"
  ],
  "abstract": "For the partial sums $(S_n)$ of independent random variables we define a stochastic process $s_n(t):=(1/d_n)\\sum_{k \\le [nt]} ({S_k}/{k}-\\mu)$ and prove that $$(1/{\\log N})\\sum_{n\\le N}(1/n)\\mathbf {I}\\left\\{s_n(t)\\le x\\right\\} \\to G_t(x)\\quad \\text{a.s.}$$ if and only if $(1/{\\log N})\\sum_{n\\le N} (1/n)\\mathbb{P}\\left(s_n(t)\\le x\\right) \\to G_t(x)$, for some sequence $(d_n)$ and distribution $G_t$. We also prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables attracted to an $\\alpha$-stable law with $\\alpha\\in (1,2]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-06-05T21:52:05.000Z",
      "abstract": "For the partial sums $(S_n)$ of independent random variables we define a stochastic process $s_n(t):=(1/d_n)\\sum_{k \\le [nt]} ({S_k}/{k}-\\mu)$ and prove that $$(1/{\\log N})\\sum_{n\\le N}(1/n)\\mathbf {I}\\bigl\\{s_n(t)\\le x\\bigr\\} \\to G_t(x)\\quad \\text{a.s.}$$ if and only if $(1/{\\log N})\\sum_{n\\le N} (1/n)\\pp\\bigl(s_n(t)\\le x\\bigr) \\to G_t(x)$, for some sequence $(d_n)$ and distribution $G_t$. We also prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables attracted to an $\\alpha$-stable law with $\\alpha\\in (1,2]$.",
      "comment": null,
      "authors": [
        "Khurelbaatar Gonchigdanzan",
        "Kamil Marcin Kosinski"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-05-19T20:23:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partial sums",
      "stable law",
      "sure functional limit theorem",
      "independent random variables",
      "positive random variables"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.1073G"
    }
  }
}