Title: Finite sample Bernstein - von Mises theorems for functionals and spectral projectors of covariance matrix,
Abstract: We demonstrate that a prior influence on the posterior distribution of
covariance matrix vanishes as sample size grows. The assumptions on a prior are
explicit and mild. The results are valid for a finite sample and admit the
dimension $p$ growing with the sample size $n$. We exploit the described fact
to derive the finite sample Bernstein - von Mises theorem for functionals of
covariance matrix (e.g. eigenvalues) and to find the posterior distribution of
the Frobenius distance between spectral projector and empirical spectral
projector. This can be useful for constructing sharp confidence sets for the
true value of the functional or for the true spectral projector.