Stein operators for variables form the third and fourth Wiener chaoses

Robert E. Gaunt

Statistics & Probability Letters, 2019, vol. 145, issue C, 118-126

Abstract: Let Z be a standard normal random variable and let Hn denote the nth Hermite polynomial. In this note, we obtain Stein equations for the random variables H3(Z) and H4(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 aZ2+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 Hn(Z), n≥5, is more challenging.

Keywords: Stein’s method; Normal distribution; Hermite polynomial; Wiener chaos; Non-central chi-square distribution (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:145:y:2019:i:c:p:118-126

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DOI: 10.1016/j.spl.2018.09.001

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