Asymptotic expansions associated with some statistical estimates in the smooth case. I. Decompositions of random variables

Let $x_1,\dots,x_n$ be a sample from a distribution $\mathbf P_\theta$ with density $f(x,\theta)$, $\theta\in\Theta$, where $\Theta$ is an open set on the real line. Let $T_n$ be a Bayessian estimate or a maximum likelihood estimate. Put
$$ \Delta_i=\frac1{\sqrt n}\sum_{j=1}^n(l_i(x_j,\theta)-\mathbf E_\theta l_i(x_1,\theta)),\quad i=1,\dots,k+1,\quad k\ge1, $$
where
$$ l_i(x,\theta)= \begin{cases} \frac{\partial^i}{\partial\theta^i}\ln f(x,\theta),&f(x,\theta)\ne0, \\ 0,&f(x,\theta)=0. \end{cases} $$
Supposing regularity conditions ($f(x,\,\cdot\,)$ has $k+2$ continuous derivatives, the moments $\mathbf E_\theta|l_i(\,\cdot\,\theta)|^{k+2}$ are uniformly bounded on compacts etc.), we obtain an expansion of the form
$$ \sqrt n(T_n-\theta)=\xi_0+\xi_1\frac1{\sqrt n}+\dots+\xi_{k-1}\biggl(\frac1{\sqrt n}\biggr)^{k-1}+\widetilde\xi_{k,n}\biggl(\frac1{\sqrt n}\biggr)^k, $$
where $\xi_\theta=\Delta_1/I(\theta)$, $I(\theta)$ is Fischer's information quantity, $\xi_i$ are polynomials in $\Delta_1,\dots,\Delta_{i+1}$,
$$ \mathbf P_\theta\{|\widetilde\xi_{k,n}|>n^\delta\}=O\bigl(n^{-\frac{k-1}2-C^\delta}\bigr) $$
for each sufficiently small $\delta>0$ uniformly on compacts. This expansion implies asymptotic expansions of $\mathbf E_\theta(\sqrt n(T_n-\theta))^m$ and $\mathbf P_\theta\{\sqrt n(T_n-\theta)<z\}$.