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Stein operators for variables form the third and fourth Wiener chaoses. (English) Zbl 1407.60036

Summary: Let \(Z\) be a standard normal random variable and let \(H_n\) denote the \(n\)th 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 \(a Z^2 + b Z + c\), \(a, b, c \in \mathbb{R}\), 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 \(H_n(Z)\), \(n \geq 5\), is more challenging.

MSC:

60F05 Central limit and other weak theorems
60G15 Gaussian processes
60H07 Stochastic calculus of variations and the Malliavin calculus

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References:

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