Is there a local limit theorem for functions of Gaussian random vectors?

Assume that $$\sqrt{n} (\boldsymbol{Z}_n - \boldsymbol{\mu}) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}(\boldsymbol{0},\Sigma)$$, as $$n\to \infty$$, for some $$\boldsymbol{\mu}\in \mathbb{R}^d$$ and $$\Sigma$$ a symmetric positive definite matrix (here, $$\stackrel{\mathcal{D}}{\longrightarrow}$$ denotes the convergence in distribution). The delta method says that for a $$\mathcal{C}^1$$ function $$h: \mathbb{R}^d \rightarrow \mathbb{R}$$, we have $$\sqrt{n} (h(\boldsymbol{Z}_n) - h(\boldsymbol{\mu})) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}(\boldsymbol{0}, \nabla h (\boldsymbol{\mu})^{\top} \Sigma \, \nabla h (\boldsymbol{\mu})), \quad \text{as } n\to \infty.$$

Now, my question is: Are there more precise results in the literature involving asymptotic expansion(s) for the DENSITY function of $$h(\boldsymbol{Z}_n)$$ (with appropriate conditions on $$h$$)? We know that $$f_{h(\boldsymbol{Z}_n)}(t) = f_{\boldsymbol{Z}_n}(h^{-1}(t)) \, \left|\frac{d}{d t} h^{-1}(t)\right|, \quad t\in \mathbb{R},$$ where the last term denotes the Jacobian of the transformation. If we let $$W\sim \mathcal{N}(\boldsymbol{0}, \nabla h (\boldsymbol{\mu})^{\top} \Sigma \, \nabla h (\boldsymbol{\mu}))$$, would it be possible to obtain a result of the form: $$\frac{f_{h(\boldsymbol{Z}_n)}(t)}{f_{W}(t)} = 1 + \frac{\text{error}_1(t)}{\sqrt{n}} + \frac{\text{error}_2(t)}{n} + ~..., \quad \text{as } n\to \infty,$$ with appropriate restrictions on $$h$$ ? I found these papers:

but they don't quite answer my question.

• Presumably $Z$ should be $Z_n$, but they don't need to have a density at all under your assumption... Commented Sep 6, 2021 at 9:17
• I understand, but there could be a class of $h$ for which we could say something about the ratio $f_{h(\boldsymbol{Z}_n)}(t) / f_{W}(t)$. In Boos (1985), the first paper linked above, his theorems 1 and 2 relate to translation and scale statistics for example. Commented Sep 6, 2021 at 9:25
• Commented Sep 6, 2021 at 22:28