Tail Approximations of Integrals of Gaussian Random Fields

Details Tail Approximations of Integrals of Gaussian Random Fields Tail Approximations of Integrals of Gaussian Random Fields

2010-06-14

arxiv.org/abs/1006.2837v2

Mathematics

Probability

Scientific paper

This paper develops asymptotic approximations of $P(\int_T e^{f(t)}dt > b)$ as $b\rightarrow \infty$ for homogeneous smooth Gaussian random field, $f$, living on a compact $d$-dimensional Jordan measurable set $T$. The integral of exponent of 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 $P(\int_\Delta e^{f(t)}dt > b)$ over a small domain $\Delta$ depending on $b$, where $mes(\Delta) \rightarrow 0$ as $b\rightarrow \infty$ and $mes(\cdot)$ is the Lebesgue measure; 2. with $\Delta$ appropriately chosen, we show that $P(\int_T e^{f(t)}dt > b) =(1+o(1)) mes(T) mes^{-1}(\Delta) P(\int_\Delta e^{f(t)}dt > b)$.

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