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(t0) u∣t0 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(t0) u + v∣t0 is a local maximum of f(t) and f(t0) > v} as v → ∞. 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.
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机译:令{f(t):t∈T}是参数空间T上的光滑高斯随机场,其中T可以是欧几里得空间的一个子集,或更一般地说,是黎曼流形。当f是非平稳的时,我们提供了一个局部最大值ℙ{f(t0)> u0t0是f(t)}的局部最大值的高度分布的一般公式。此外,我们建立局部最大值ℙ{f(t0)> u + v em> ∣ t em> 0的局部超调分布的渐近逼近0是 f的局部最大值 em>( t em>)和 f em>( t em> 0)> v em>}作为 v < / em>→∞。进一步假设 f em>是各向同性的,当 T em>是欧几里得或任意维数球体时,我们应用与高斯正交系相关的随机矩阵理论的技术来显式地计算这种条件概率。 。此类计算是由统计问题(在存在平滑高斯噪声的情况下检测峰值)推动的。
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