  Normal comparison lemma and Slepian's inequality are essential tools in the
study of Gaussian processes. In this paper we extend normal comparison lemma
and derive various related comparison inequalities including Slepian's
inequality for order statistics of two Gaussian arrays.The derived results can
be applied in numerous problems related to the study of the supremum of order
statistics of Gaussian processes. In order to illustrate the range of possible
applications, we analyze the lower tail behaviour of order statistics of
self-similar Gaussian processes and derive mixed Gumbel limit theorems.
