An asymptotic expansion for distributions of sums of a special form with application to minimum contrast estimates

Let $\mathbf{Y}_i=(Y_{i1},\dots,Y_{ik})$, $i=1,\dots,n$, be independent identically distributed random vectors in $R^k$ and $\sum_{jn}=\sum_{i=1}^n Y_{ij}$, $j=1,\dots,k$, $k\ge 3$. Put
$$ f(\mathbf{t})=\mathbf{E}\exp [i(t_1 Y_{11}+\dots+t_k Y_{1k})], \qquad \mathbf{t}=(t_1,\dots,t_k)\in R^k. $$
Let there be given some numbers $C>0$ and $\alpha_j>1$, $j=2,\dots,k$, and sequence $\{z_{jn}\}$, $j=2,\dots,k$, such that $n^{-j/2}|z_{jn}|\leq Cn^{-\alpha_j/2}$. Let
$$ T_n=n^{-1/2}\Sigma_{1n}+n^{-1}z_{2n}\Sigma_{2n}+\dots+n^{-k/2}z_{kn}\Sigma_{kn}. $$

Theorem 1. \textit{Suppose that $\mathrm{(a)}$ $\mathbf{E}|Y_{ij}|^{k/\alpha_j}<\infty$, $j=1,\dots,k$ (putting $\alpha_1=1$); $\mathrm{(b)}$ $\mathbf{E}Y_{1j}=0$ for those $j\in \{1,\ldots,k\}$ for which $k/\alpha_j \geq 1$; $\mathrm{(c)}\sup_{||\mathbf{t}||>\delta,\,\mathbf{t}\in R^*}|f(t)|<1$ for any $\delta>0$, where $R^*=\{\mathbf{t}\in R^k:t_j=0$ whenever $\alpha_j>k-2,\,j=2,\dots,k\}$. Without loss of generality, assume that $\mathbf{E}Y_{11}^2=1$. Then there exist polynomials $P_m(y,z_2,\dots,z_k)$, $m=1,\dots,k-1$, with coefficients dependent on the moments $\mathbf{E}(Y_{11}^{h_1}\dots Y_{1k}^{h_k})$ with $h_j\geq 0$, $\sum_{j=1}^k \alpha_j h_j\leq k$, such that}
$$ \sup_y \biggl|\mathbf{P}\{T_n<y\}-\biggl[\Phi(y)+\sum_{m=1}^{k-2}n^{-m/2}P_m(y,z_{2n},\dots,z_{kn})\varphi(y)\biggr]\biggr|=o\bigl(n^{-\frac{k-2}{2}}\bigr), $$
$\Phi$ and $\varphi$ being the standart normal distribution function and density.
Using the theorem, asymptotic expansions for the distributions of minimum contrast estimates for a one-dimensional parameter are obtained. A short formulation of this latter result was given in this journal, XVII, 2 (1972), 387–388 (Theorem 2).