An asymptotic expansion for the distribution of a statistic admitting an asymptotic expansion

Let $\mathbf{X}_i$, $i=1,\dots,n$ be $p$-dimensional independent identically distributed random vectors and $\mathbf{S}_n=n^{-1/2}\sum\mathbf{X}_i$. Let $\mathbf{H}_0(\mathbf{x})$ be a linear function from $R^p$ into $R^s$, $s\leq p$, and $\mathbf{H}_j(\mathbf{x})=(H_{j1}(\mathbf{x}),\dots,H_{js}(\mathbf{x}))$, $j=1,\dots,k$, where $\mathbf{H}_{jl}(\mathbf{x})$, $\mathbf{x}\in R^p$, $l=1,\dots,s$, are polinomials. For the distribution of
\begin{equation} \mathbf{Z}_n=\mathbf{H}_0(\mathbf{S}_n)+\sum_{j=1}^k n^{-j/2}\mathbf{H}_j(\mathbf{S}_n) \tag{1} \end{equation}
an asymptotic expansion of the Edgeworth type is obtained. A modification of this result is given for the case when the right hand side of (1) contains a remainder term converging to zero at a certain rate.