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.