## Abstract

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→∞$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.

Original language | English (US) |
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Pages (from-to) | 213-240 |

Number of pages | 28 |

Journal | Extremes |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - Dec 11 2015 |

Externally published | Yes |

## Keywords

- Euler characteristic
- Gaussian orthogonal ensemble
- Height
- Isotropic field
- Local maxima
- Overshoot
- Riemannian manifold
- Sphere

## ASJC Scopus subject areas

- Statistics and Probability
- Engineering (miscellaneous)
- Economics, Econometrics and Finance (miscellaneous)