Abstract Integral equations of the type in Bernard and Shipley (1972) are used to calculate approximately the distribution of the first time T a that a Gaussian process X ( t ) crosses a threshold a from below, referred as the first passage time. These equations involve kernels which are currently calculated numerically from multidimensional integrals. It is shown that Slepian models can be used to calculate efficiently the kernels of the integral equations for the distribution of T a . The construction of Slepian-based kernels is illustrated by numerous examples. Estimates of the distribution of T a are compared to those by Monte Carlo and mean crossing rates for broad- and narrow-band Gaussian processes.

中文翻译：

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Slepian 模型对高斯过程的首次通过时间

摘要 Bernard 和 Shipley (1972) 中的积分方程用于近似计算高斯过程 X ( t ) 从下方越过阈值 a 的第一时间 T a 的分布，称为第一次通过时间。这些方程涉及当前从多维积分进行数值计算的核。结果表明，Slepian 模型可用于有效计算 T a 分布的积分方程的核。许多示例说明了基于 Slepian 的内核的构造。将 T a 的分布估计值与 Monte Carlo 估计值和宽带和窄带高斯过程的平均交叉率进行比较。