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I am reading a paper (https://arxiv.org/abs/1001.3448) and they mentioned Stein's lemma (below) as a useful fact without proof, I also read the reference in the paper but I got nothing. Please help me any material contained proof of this lemma.

(Stein's lemma) For jointly Gaussian variables $Z_1, Z_2$ with zero mean and for any function $\psi: \mathbb{R} \rightarrow \mathbb{R}$, where $\mathbb{E}\{\psi'(Z_1)\}, \mathbb{E}\{\psi'(Z_2)\}$ exist, the following holds $$\mathbb{E}\{Z_1\psi(Z_2)\} = \text{Cov}(Z_1,Z_2)\mathbb{E}\{\psi'(Z_2)\}$$

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You can write $Z_1=\alpha Z_2+\tilde{Z}_1$ with $Z_2,\tilde{Z}_1$ are independent. Then $$\mathbb{E}(Z_1\psi(Z_2))=\alpha\mathbb{E}(Z_2\psi(Z_2))=\frac{\alpha}{\sqrt{2\pi\sigma^2}}\int_{\mathbb{R}}xe^{x^2/2\sigma^2}\psi(x)dx $$ one can conclude by integration by parts.

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  • $\begingroup$ And why we can write $Z_1 = \alpha Z_2 + \tilde{Z}_1$ with $Z_2, \tilde{Z}_1$ are independent? $\endgroup$
    – Quicky2357
    Commented May 16, 2020 at 5:53
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    $\begingroup$ I have my own answer now, I choose $\alpha = \frac{\mathbb{E}\{Z_1Z_2\}}{\mathbb{E}\{Z_2^2\}}$ $\endgroup$
    – Quicky2357
    Commented May 17, 2020 at 4:23

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