Bernstein - von Mises Theorem for growing parameter dimension

This paper revisits the prominent Fisher, Wilks, and Bernstein – von Mises (BvM) results from different viewpoints. Particular issues to address are: nonasymptotic framework with just one finite sample, possible model misspecification, and a large parameter dimension. In particular, in the case of an i.i.d. sample, the mentioned results can be stated for any smooth parametric family provided that the dimension $p$ of the parameter space satisfies the condition “$p^2\mathrm{/}n$ is small” for the Fisher expansion, while the Wilks and the BvM results require “$p^3\mathrm{/}n$ is small”.