## Abstract

Let X = {X(t), t ϵ R^{N}} be a centered Gaussian random field with stationary increments and X(0) = 0. For any compact rectangle T ⊂ R^{N} and u ϵ R, denote by A_{u} = {t ϵ T :X(t) ≥ u} the excursion set. Under X(·) ∈ C^{2}(R^{N}) and certain regularity conditions, the mean Euler characteristic of A_{u}, denoted by E{ø(A_{u})}, is derived. By applying the Rice method, it is shown that, as u→∞, the excursion probability P{suptϵT X(t) ≥ u} can be approximated by E{ø(A_{u})} such that the error is exponentially smaller than E{ø(A_{u})}. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.

Original language | English (US) |
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Pages (from-to) | 722-759 |

Number of pages | 38 |

Journal | Annals of Applied Probability |

Volume | 26 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2016 |

Externally published | Yes |

## Keywords

- Euler characteristic
- Excursion probability
- Excursion set
- Gaussian random fields with stationary increments
- Super-exponentially small

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty