On sequential estimation under the conditions of the local asymptotic normality

Let $\mathscr E_n=\{\mathbf P_{n,\theta};\theta\in\Theta\}$ be a sequence of families of probability measures and $l(x)$ be a loss function such that $l(x)=l(-x)$, $l(x)\ge l(z)$ if $|x|\ge|z|$,
$$ \psi(\mu)=\int l(x/\mu)e^{-x^2/2}\,dx<\infty\qquad\text{and}\qquad\psi(\mu)\to 0,\quad \mu\to\infty. $$

Theorem 1. {\it Let $\mathscr E_n$ be locally asymptotically normal at the point $t\in\Theta$ and $(\{T_m^{(n)}\},\tau_n)$ be a sequence of sequential estimation procedures. Then for arbitrary positive constants $\gamma$, $\varepsilon$, $\delta$ there exist $b_0(\gamma,\varepsilon,\delta)$ and $n_0(\gamma,\varepsilon,\delta,b)$ such that for $b\ge b_0(\gamma,\varepsilon,\delta)$ and $n\ge n_0(\gamma,\varepsilon,\delta,b)$ inequality (2) is valid.}
As a consequence of this theorem we show that if $\tilde l(1/\mu)$ is convex function of $\mu$ and (3) holds, then the inequality (4) is valid. Asymptotical normality and local asymptotical minimax properties of maximum likelihood, Bayes and generalized Bayes estimates are established.