Let
- $n\in\mathbb N$ with $n>1$
- $f\in C^3(\mathbb R)$ with $f>0$ and $$I:=\int\frac{{f'(x)}^2}{f(x)}\:{\rm d}x<\infty$$
- $g:=\ln f$ (assume $g'$ is Lipschitz continuous), $$r_n(x):=\frac1{n-1}\sum_{i=2}^n{g'(x_i)}^2\;\;\;\text{for }x\in\mathbb R^n$$ and $$F_n:=\left\{x\in\mathbb R^n:|r_n(x)-I|<d^{-1/8}\right\}$$
- $\ell>0$ and $\sigma_n^2:=\ell^2/(n-1)$
- $Y_1\sim\mathcal N_{x_1,\:\sigma_n^2}$
- $\varphi\in C_c^\infty(\mathbb R)$
- $\Phi$ denote the cumulative distribution function of $\mathcal N_{0,\:1}$ and $$\Psi_x(y_1):=\Phi\left(\frac{g(y_1)-g(x_1)}{\ell\sqrt{r_n(x)}}-\frac\ell2\sqrt{r_n(x)}\right)+e^{g(y_1)-g(x_1)}\Phi\left(-\frac{g(y_1)-g(x_1)}{\ell\sqrt{r_n(x)}}-\frac\ell2\sqrt{r_n(x)}\right)$$ for $x\in\mathbb R^n$ and $y_1\in\mathbb R$
I want to show that $$n\operatorname E\left[(\varphi(Y_1)-\varphi(x_1))\Psi_x(Y_1)\right]=n(\varphi'(x_1)g'(x_1)+\varphi''(x_1))\Phi\left(-\frac\ell2\sqrt{r_n(x)}\right)\sigma_n^2+\mathcal O(n^{-1/2})\tag1$$ uniformly with respect to $x\in F_n$.
The idea is to use Taylor's theorem to obtain $$\varphi(Y_1)-\varphi(x_1)=\varphi'(x_1)(Y_1-x_1)+\frac12\varphi''(x_1)(Y_1-x_1)^2+\frac16\varphi'''(Z_1)(Y_1-x_1)^3\tag2$$ and $$\Psi_x(Y_1)=\Psi_x(x_1)+\Psi_x'(x_1)(Y_1-x_1)+\frac12\Psi_x''(W_1)(Y_1-x_1)^2=(2+g'(x_1)(Y_1-x_1))\Phi\left(-\frac\ell2\sqrt{r_n(x)}\right)+\frac12\Psi_x''(W_1)(Y_1-x_1)^2\tag3$$ for some random variables $Z_1,W_1\in[x_1\wedge Y_1,x_1\vee Y_1]$. This yields that the left-hand side of $(1)$ is equal to $$n\left((\varphi'(x_1)g'(x_1)+\varphi''(x_1))\Phi\left(-\frac\ell2\sqrt{r_n(x)}\right)\sigma_n^2+\color{red}{\frac1 \Phi\left(-\frac\ell2\sqrt{r_n(x)}\right)\operatorname E\left[\varphi'''(Z_1)(Y_1-x_1)^3\right]+\frac12\varphi'(x_1)\operatorname E\left[\Psi_x''(W_1)(Y_1-x_1)^3\right]+\frac14\varphi''(x_1)\operatorname E\left[\Psi_x''(W_1)(Y_1-x_1)^4\right]+\frac16g'(x_1)\Phi\left(-\frac\ell2\sqrt{r_n(x)}\right)\operatorname E\left[\varphi'''(Z_1)(Y_1-x_1)^4\right]+\frac1{12}\operatorname E\left[\varphi'''(Z_1)\Psi_x''(W_1)(Y_1-x_1)^5\right]}\right)$$
We need to show that the red term is $\mathcal O(n^{-3/2})$. While $\varphi$ and all its derivatives are bounded on a common compact set $K$, I don't know how I need to deal with the $\Psi_x''(W_1)$ inside the expectations.
The claim can be found at the end of page 8 / beginning of page 9 in this paper: https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254