Tail approximations of integrals of Gaussian random fields

• Autores: Jingchen Liu
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 40, Nº. 3, 2012, págs. 1069-1104
• Idioma: inglés
• DOI: 10.1214/10-AOP639
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• Resumen

  • This paper develops asymptotic approximations of P(∫Tef(t) dt > b) as b → ∞ for a homogeneous smooth Gaussian random field, f, living on a compact d-dimensional Jordan measurable set T. The integral of an exponent of a Gaussian random field is an important random variable for many generic models in spatial point processes, portfolio risk analysis, asset pricing and so forth.

    The analysis technique consists of two steps: 1. evaluate the tail probability P(∫Ξef(t) dt > b) over a small domain Ξ depending on b, where mes(Ξ) → 0 as b → ∞ and mes(⋅) is the Lebesgue measure; 2. with Ξ appropriately chosen, we show that P(∫Tef(t) dt > b) = (1 + o(1)) mes(T) mes−1(Ξ) P(∫Ξef(t) dt > b).