TAIL APPROXIMATIONS OF INTEGRALS OF GAUSSIAN RANDOM FIELDS

• 成果类型: Article
• 署名作者: Liu, Jingchen
• 署名单位: Columbia University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP639
• 发表日期: 2012
• 页码: 1069-1104
• 关键词: value-at-risk; time-series; REGRESSION-MODEL; random-variables; brownian-motion; maximum; counts; INEQUALITY; sums

摘要

This paper develops asymptotic approximations of P(f(T) e(f(t)) dt > b) as b -> infinity 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(f(Xi) e(f(t)) dt > b) over a small domain Xi depending on b, where mes(Xi) -> 0 as b -> infinity and mes(.) is the Lebesgue measure; 2. with Xi appropriately chosen, we show that P(f(T) e(f(t)) dt > b) = (1 + o(1)) mes(T) x mes(-1)(Xi)P(f(Xi) e(f(t)) dt > b).