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 language | English |
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Pages (from-to) | 118-126 |
Number of pages | 9 |
Journal | Statistics & Probability Letters |
Volume | 145 |
Early online date | 14 Sept 2018 |
DOIs | |
Publication status | Published - 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