Critical dimension in semiparametric Bernstein - von Mises Theorem

The classical Bernstein-von Mises (BvM) result is considered in finite sample non-asymptotic setup. The approach follows the idea of Spokoiny (2012) and it is mainly based on the local majorization device which is introduced in that paper. The main attention is paid to notion of critical dimension which determines maximum allowed problem dimension ($p$) given sample size ($n$). It is shown that condition "($p^3 / n$) is small" is sufficient for BvM result to be valid. Also we provide an example which shows the failure of BvM result in case, when this condition is not satisfied.