On the asymptotic properties of Bayes-type estimators with general loss functions

We study the asymptotic behavior of Bayes-type estimators and give sufficient conditions to obtain the asymptotic limit distribution of the estimation error. We assume so called polynomial-type large deviation inequalities and prove the asymptotic equivalence of the estimation errors of Bayes-type and M-estimators by virtue of Ibragimov-Has'minskiÄ­ theory. The results can be applied to several cases such as diffusion processes and jump diffusion processes. In this paper, we focus on the application to ergodic diffusion processes and ergodic jump diffusion processes, demonstrating asymptotic normality and convergence of moments for Bayes-type estimators with general loss functions.