Extreme Values and High Boundary Crossings of Locally Stationary Gaussian Processes

We consider the large values of a locally stationary Gaussian process which satisfies Berman's condition on the long range dependence. We present some limit results on the exceedances of the process above a certain general smooth high boundary. This allows deriving the limiting distribution of the maximum up to time T, for example, in the case of a standardized process with a constant boundary or in the case of a nonstandardized process with a smooth trend