Stein operators for variables form the third and fourth Wiener chaoses

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Abstract

Let Z be a standard normal random variable and let H n denote the nth Hermite polynomial. In this note, we obtain Stein equations for the random variables H 3 (Z) and H 4 (Z), which represent a first step towards developing Stein's method for distributional limits from the third and fourth Wiener chaoses. Perhaps surprisingly, these Stein equations are fifth and third order linear ordinary differential equations, respectively. As a warm up, we obtain a Stein equation for the random variable aZ 2 +bZ+c, a,b,c∈R, which leads us to a Stein equation for the non-central chi-square distribution. We also provide a discussion as to why obtaining Stein equations for H n (Z), n≥5, is more challenging.

Original languageEnglish
Pages (from-to)118-126
Number of pages9
JournalStatistics & Probability Letters
Volume145
Early online date14 Sept 2018
DOIs
Publication statusPublished - 1 Feb 2019

Keywords

  • Stein’s methodNormal distributionHermite polynomialWiener chaosNon-central chi-square distribution
  • Hermite polynomial
  • Wiener chaos
  • Normal distribution
  • Stein's method
  • Non-central chi-square distribution

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