  In this paper, by using the exact tail asymptotics derived by Debicki,
Hashorva and Ji (Ann. Probab. 2014), we proved the Gumbel limit theorem for the
maximum of a class of non-homogeneous Gaussian random fields. By using the
obtained results, we also derived the Gumbel laws for Shepp statistics of
fractional Brownian motion and Gaussian integrated process as well as the
Gumbel law for Storage process with fractional Brownian motion as input.
