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In the paper "Spin glasses and Stein's method" (https://arxiv.org/pdf/0706.3500.pdf), Sourav Chatterjee established Stein's equation for mixtures of two Gaussian densities in $\mathbb{R}$, which takes the form as in page 15 of https://statweb.stanford.edu/~souravc/talk_spin.pdf.

Is this form extendable to mixtures of two Gaussian densities in $\mathbb{R}^d$?

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The answer is yes. In general, if a random variable $X$ has a Lebesgue density $ f_X : \mathbb{R}^d \to \mathbb{R}_+ $, a Stein operator is given by $ g \mapsto \frac{\mathrm{div}(g f_X) }{ f_X } $, see e.g. https://people.maths.bris.ac.uk/~mb13434/prst_talks/Y_Swan_PrSt_Bristol_141107.pdf page 26 for $d = 1$ and page 47 for the multivariate case.

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