Asymptotic expansion for the distribution of a statistic admitting a stochastic expansion. I

Let $(Y_{0i},\mathbf Y_i)=(Y_{0i},Y_{1i},\dots,Y_{pi})$, $i=1,\dots,n$, be i.i.d. random vectors in $R^{p+1}$, and $\{h_j\}$ be a finite set of polinomials of $p+1$ variables. Let
\begin{gather*} S_n=n^{-1/2}\sum Y_{0i},\qquad T_{nl}=n^{-1/2}\sum Y_{li},\qquad\mathbf T_n=(T_{n1},\dots,T_{np}), \\ Z_n=S_n+\sum n^{-j/2}h_j(S_n,\mathbf T_n). \end{gather*}
In the paper an asymptotic expansion of the Edgeworth's type for the distribution function of $Z_n$ is obtained under conditions which are weaker than those previously known.