I'm reading this paper and on page 8 the authors state without proof an asymptotic expansion of a multivariate Gaussian integral in terms of the covariance obtained by applying what they call the "Lebesgue theorem" (my guess is that they mean the Riemann-Lebesgue lemma).

Reformulated, the problem is as follows: Let $X_n, Y_n$ be standard Gaussians such that their covariance $A(n)=\operatorname{E} (X_nY_n)$ tends to zero polynomially fast as $n \to \infty$. Furthermore, let $g$ be an even function with polynomial growth, such that $\operatorname{E}g(X_n)=0$. I would like to understand why

$$\operatorname{E} \left( g(X_n) g(Y_n) \right) \sim \operatorname{E} \left( g(X)g(Y) (X^2Y^2 - X^2 - Y^2 +1 ) \right) \frac{1}{2} A^2(n)$$

as $n \to \infty$, where $X, Y$ are two independent standard Gaussians.

Can somebody please hint me in the right direction?

Thanks a lot!