### Abstract

In this paper, we consider the extreme behavior of a Gaussian random field f (t) living on a compact set T. In particular, we are interested in tail events associated with the integral Σ_{T} e^{f(t)} dt.We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field f (given that Σ_{T} e^{f(t)} dt exceeds a large value) in total variation. Based on this approximation, we show that the tail event of Σ_{T} e^{f(t)} dt is asymptotically equivalent to the tail event of sup_{T} γ^{(t)} where γ^{(t)} is a Gaussian process and it is an affine function of f (t) and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of log b to compute the probability P(Σ_{T} e^{f(t)} dt > b) with a prescribed relative accuracy.

Original language | English (US) |
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Pages (from-to) | 1691-1738 |

Number of pages | 48 |

Journal | Annals of Applied Probability |

Volume | 24 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2014 |

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### Keywords

- Change of measure
- Efficient simulation
- Gaussian process

### Cite this

*Annals of Applied Probability*,

*24*(4), 1691-1738. https://doi.org/10.1214/13-AAP960