Extremes of a Type of Locally Stationary Gaussian Random Fields with Applications to Shepp Statistics

Let $$\{Z(\tau ,s), (\tau ,s)\in [a,b]\times [0,T]\}$$ { Z ( τ , s ) , ( τ , s ) ∈ [ a , b ] × [ 0 , T ] } with some positive constants a, b, T be a centered Gaussian random field with variance function $$\sigma ^{2}(\tau ,s)$$ σ 2 ( τ , s ) satisfying $$\sigma ^{2}(\tau ,s)=\sigma ^{2}(\tau )$$ σ 2 ( τ , s ) = σ 2 ( τ ) . We first derive the exact tail asymptotics (as $$u \rightarrow \infty $$ u → ∞ ) for the probability that the maximum $$M_H(T) = \max _{(\tau , s) \in [a, b] \times [0, T]} [Z(\tau , s) / \sigma (\tau )]$$ M H ( T ) = max ( τ , s ) ∈ [ a , b ] × [ 0 , T ] [ Z ( τ , s ) / σ ( τ ) ] exceeds a given level u, for any fixed $$0 0$$ T > 0 ; and we further derive the extreme limit law for $$M_{H}(T)$$ M H ( T ) . As applications of the main results, we derive the exact tail asymptotics and the extreme limit laws for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as inputs.