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 $$0< a< b < \infty $$ 0 < a < b < ∞ and $$T > 0$$ 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.

中文翻译：

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一类局部平稳高斯随机场的极值及其在 Shepp 统计中的应用

让 $$\{Z(\tau ,s), (\tau ,s)\in [a,b]\times [0,T]\}$$ { Z ( τ , s ) , ( τ , s ) ∈ [ a , b ] × [ 0 , T ] } 有一些正常数 a , b , T 是一个中心高斯随机场，方差函数 $$\sigma ^{2}(\tau ,s)$$ σ 2 ( τ , s ) 满足 $$\sigma ^{2}(\tau ,s)=\sigma ^{2}(\tau )$$ σ 2 ( τ , s ) = σ 2 ( τ ) 。我们首先为最大 $$M_H(T) = \max _{(\tau , s) \in [a, b] \times [0, T]} [Z(\tau , s) / \sigma (\tau )]$$ MH ( T ) = max ( τ , s ) ∈ [ a , b ] × [ 0 , T ] [ Z ( τ , s ) / σ ( τ ) ] 超过给定水平 u ，对于任何固定的 $$0< a< b < \infty $$ 0 < a < b < ∞ 和 $$T > 0$$ T > 0; 我们进一步推导出 $$M_{H}(T)$$ MH ( T ) 的极限定律。作为主要结果的应用，