Asymptotic Expansion Under Degeneracy

Abstract

We will consider a stochastic expansion described by random variables whose covariance matrix is asymptotically degenerate. Though the conventional approach with Bhattacharya-Ghosh's transform requires the nondegeneracy of the covariance matrix, it is known that this method still works even in degenerate cases with the help of the so-called global approach. In this paper, we explain this fact and also mention, as an example, the third order asymptotic expansion of the maximum likelihood estimator for the O-U process.