# pr.probability – 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 $$nto infty$$, 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
$$sqrt{n} (h(boldsymbol{Z}_n) – h(boldsymbol{mu})) stackrel{mathcal{D}}{longrightarrow} mathcal{N}(boldsymbol{0}, nabla h (boldsymbol{mu})^{top} Sigma , nabla h (boldsymbol{mu})), quad text{as } nto infty.$$

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
$$f_{h(boldsymbol{Z}_n)}(t) = f_{boldsymbol{Z}_n}(h^{-1}(t)) , left|frac{d}{d t} h^{-1}(t)right|, quad tin mathbb{R},$$
where the last term denotes the Jacobian of the transformation.
If we let $$Wsim mathcal{N}(boldsymbol{0}, nabla h (boldsymbol{mu})^{top} Sigma , nabla h (boldsymbol{mu}))$$, would it be possible to obtain a result of the form:
$$frac{f_{h(boldsymbol{Z}_n)}(t)}{f_{W}(t)} = 1 + frac{text{error}_1(t)}{sqrt{n}} + frac{text{error}_2(t)}{n} + ~…, quad text{as } nto infty,$$
with appropriate restrictions on $$h$$ ?
I found these papers:

but they don’t quite answer my question.

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