Let {f(t) : t ∈ T} be a smooth Gaussian random field over a parameter space T, where T may be a subset of Euclidean space or, more generally, a Riemannian manifold. We provide a general formula for the distribution of the height of a local maximum ℙ{f(t ₀) > u|t ₀ is a local maximum of f(t)} when f is non-stationary. Moreover, we establish asymptotic approximations for the overshoot distribution of a local maximum ℙ{f(t ₀) > u+v|t ₀ is a local maximum of f(t) and f(t ₀) > v} as \(v\to \infty \). Assuming further that f is isotropic, we apply techniques from random matrix theory related to the Gaussian orthogonal ensemble to compute such conditional probabilities explicitly when T is Euclidean or a sphere of arbitrary dimension. Such calculations are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.

中文翻译：

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高斯随机场局部极大值高度的分布。


令 { f ( t ) : t ∈ T } 为参数空间T上的平滑高斯随机场，其中T可以是欧几里得空间的子集，或更一般地，黎曼流形。我们提供了局部最大值高度分布的通用公式 ℙ{ f ( t ₀ ) > u |当f非平稳时， t ₀是f ( t )} 的局部最大值。此外，我们建立了局部最大值 ℙ{ f ( t ₀ ) > u + v | | 的超调分布的渐近近似。 t ₀是f ( t ) 和f ( t ₀ ) > v } 的局部最大值，即\(v\to \infty \) 。进一步假设f是各向同性的，我们应用与高斯正交系综相关的随机矩阵理论中的技术来显式计算当T是欧几里得或任意维度的球体时的条件概率。此类计算的动机是在存在平滑高斯噪声的情况下检测峰值的统计问题。