ON THE EQUIVALENCE OF THE TUBE AND EULER CHARACTERISTIC METHODS FOR THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS OVER PIECEWISE SMOOTH DOMAINS
ON THE EQUIVALENCE OF THE TUBE AND EULER CHARACTERISTIC METHODS FOR THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS OVER PIECEWISE SMOOTH DOMAINS
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DOI：
10.1214/aoap/1026915624
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发表时间：
2002-05
影响因子：
1.8
通讯作者：
A. Takemura;S. Kuriki
中科院分区：
文献类型：
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作者：
A. Takemura;S. Kuriki
考虑具有有限 Karhunen--Loeve 展开形式 $Z(u) = \sum_{i=1}^n u_i z_i$ 的高斯随机场，其中 $z_i$, $i=1,\ldots,n,$ 是独立的标准正态变量，$u=(u_1,\ldots,u_n)'$ 范围在索引集 $M$ 上，索引集 $M$ 是单位球面的子集$R^n$ 中的 $S^{n-1}$。在非常普遍的假设$M$是具有分段平滑边界的流形的情况下，我们证明了两种当前可用的获得$Z(u)$最大值尾部概率渐近展开的方法的有效性和等价性。一种是管法，其中评估索引集 $M$ 周围的管的体积。另一种是欧拉特征法，评估偏移集的欧拉特征的期望。 R. J. Adler 最近的一篇论文对这种等价性进行了一般性讨论。为了证明其等价性，我们证明了具有分段平滑边界的流形的莫尔斯定理的一个版本。
Consider a Gaussian random field with a finite Karhunen--Loeve expansion of the form $Z(u) = \sum_{i=1}^n u_i z_i$, where $z_i$, $i=1,\ldots,n,$ are independent standard normal variables and $u=(u_1,\ldots,u_n)'$ ranges over an index set $M$, which is a subset of the unit sphere $S^{n-1}$ in $R^n$. Under a very general assumption that $M$ is a manifold with a piecewise smooth boundary, we prove the validity and the equivalence of two currently available methods for obtaining the asymptotic expansion of the tail probability of the maximum of $Z(u)$. One is the tube method, where the volume of the tube around the index set $M$ is evaluated. The other is the Euler characteristic method, where the expectation for the Euler characteristic of the excursion set is evaluated. General discussion on this equivalence was given in a recent paper by R. J. Adler. In order to show the equivalence we prove a version of the Morse theorem for a manifold with a piecewise smooth boundary.