{
  "id": "math/9309211",
  "version": "v2",
  "published": "1993-09-13T16:56:47.000Z",
  "updated": "1999-12-06T14:53:53.000Z",
  "title": "Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics",
  "authors": [
    "Victor H. de la Peña",
    "Stephen J. Montgomery-Smith"
  ],
  "journal": "Annals Prob. 23, (1995), 806-816",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper the following result, which allows one to decouple U-Statistics in tail probability, is proved in full generality. Theorem 1. Let $X_i$ be a sequence of independent random variables taking values in a measure space $S$, and let $f_{i_1...i_k}$ be measurable functions from $S^k$ to a Banach space $B$. Let $(X_i^{(j)})$ be independent copies of $(X_i)$. The following inequality holds for all $t \\ge 0$ and all $n\\ge 2$, $$ P(||\\sum_{1\\le i_1 \\ne ... \\ne i_k \\le n} f_{i_1 ... i_k}(X_{i_1},...,X_{i_k}) || \\ge t) \\qquad\\qquad$$ $$ \\qquad\\qquad\\le C_k P(C_k||\\sum_{1\\le i_1 \\ne ... \\ne i_k \\le n} f_{i_1 ... i_k}(X_{i_1}^{(1)},...,X_{i_k}^{(k)}) || \\ge t) .$$ Furthermore, the reverse inequality also holds in the case that the functions $\\{f_{i_1... i_k}\\}$ satisfy the symmetry condition $$ f_{i_1 ... i_k}(X_{i_1},...,X_{i_k}) = f_{i_{\\pi(1)} ... i_{\\pi(k)}}(X_{i_{\\pi(1)}},...,X_{i_{\\pi(k)}}) $$ for all permutations $\\pi$ of $\\{1,...,k\\}$. Note that the expression $i_1 \\ne ... \\ne i_k$ means that $i_r \\ne i_s$ for $r\\ne s$. Also, $C_k$ is a constant that depends only on $k$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "1999-12-06T14:53:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tail probability",
      "multivariate u-statistics",
      "decoupling inequalities",
      "independent random variables",
      "decouple u-statistics"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1993math......9211D"
    }
  }
}