Asymptotical behaviour of some statistical estimators. II. Limiting theorems for the a posteriory density and Bayesian estimators

In the second part of the paper we use propositions, methods and results of the first part appeared in the previous issue of this journal.
Under conditions I–IV of § 1, we prove theorems about behaviour of the a posteriory density (similar to the well-known Le Cam's results [2]), Bayesian estimators $t_n^{(a)}$ for the risk function $\|\theta\|^a$, Pitman's estimators of the location parameter etc. We prove, for example, that the estimators $t_n^{(a)}$, for different $a\ge1$, are equivalent in the sense that
$$\mathbf E\{\sqrt n\bigl|t_n^{(a_1)}-t_n^{(a_2)}\bigr|\}^p\underset{n\to\infty}\longrightarrow0\quad(p>0).$$