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Rare-event simulation and efficient discretization for the supremum of Gaussian random fields

Part of: Nonparametric inference Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  21 March 2016

Xiaoou Li  and
Jingchen Liu
Xiaoou Li*
Affiliation:
Columbia University
Jingchen Liu*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA.
∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA.
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Abstract

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In this paper we consider a classic problem concerning the high excursion probabilities of a Gaussian random field f living on a compact set T. We develop efficient computational methods for the tail probabilities ℙ{supTf(t) > b}. For each positive ε, we present Monte Carlo algorithms that run in constant time and compute the probabilities with relative error ε for arbitrarily large b. The efficiency results are applicable to a large class of Hölder continuous Gaussian random fields. Besides computations, the change of measure and its analysis techniques have several theoretical and practical indications in the asymptotic analysis of Gaussian random fields.

Keywords

Gaussian random fieldhigh-level excursionMonte Carlotail distributionefficiency

MSC classification

Primary: 60G15: Gaussian processes 65C05: Monte Carlo methods
Secondary: 60G60: Random fields 62G32: Statistics of extreme values; tail inference
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 787 - 816
Copyright
Copyright © Applied Probability Trust 2015 

References

Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.Google Scholar
Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
Adler, R. J., Blanchet, J. H. and Liu, J. (2012). Efficient Monte Carlo for high excursions of Gaussian random fields. Ann. Appl. Prob. 22, 1167–1214.Google Scholar
Adler, R. J., Samorodnitsky, G. and Taylor, J. E. (2013). High level excursion set geometry for non-Gaussian infinitely divisible random fields. Ann. Prob. 41, 134–169.Google Scholar
Aldous, D. J. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.Google Scholar
Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google Scholar
Azaïs, J.-M. and Wschebor, M. (2008). A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. Stoch. Process. Appl. 118, 1190–1218. (Erratum: 120 (2010), 2100–2101.)Google Scholar
Azais, J. M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken, NJ.Google Scholar
Berman, S. M. (1972). Maximum and high level excursion of a Gaussian process with stationary increments. Ann. Math. Statist. 43, 1247–1266.Google Scholar
Berman, S. M. (1985). An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. J. Appl. Prob. 22, 454–460.Google Scholar
Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30, 207–216.Google Scholar
Borell, C. (2003). The Ehrhard inequality. C. R. Math. Acad. Sci. Paris 337, 663–666.CrossRefGoogle Scholar
Bucklew, J. A. (2004). Introduction to Rare Event Simulation. Springer, New York.Google Scholar
Cirel'son, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian sample functions. In Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975), Springer, Berlin, pp. 20–41.Google Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.Google Scholar
Dudley, R. M. (2010). Sample functions of the Gaussian process. In Selected Works of R. M. Dudley, Springer, New York, pp. 187–224.Google Scholar
Juneja, S. and Shahabuddin, P. (2006). Rare event simulation techniques. In Handbooks in Operations Research and Management Science: Simulation, North-Holland, Amsterdam, pp. 291–350.Google Scholar
Landau, H. J. and Shepp, L. A. (1970). Supremum of a Gaussian process. Sankhyā Ser. A 32, 369–378.Google Scholar
Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin.CrossRefGoogle Scholar
Li, X. and Liu, J. (2013). Rare-event simulation and efficient discretization for the Supremum of Gaussian random fields. Available at http://arxiv.org/abs/1309.7365.Google Scholar
Liu, J. (2012). Tail approximations of integrals of Gaussian random fields. Ann. Prob. 40, 1069–1104.Google Scholar
Liu, J. and Xu, G. (2012). Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes. Ann. Statist. 40, 262–293.Google Scholar
Liu, J. and Xu, G. (2013). On the density functions of integrals of Gaussian random fields. Adv. Appl. Prob. 45, 398–424.CrossRefGoogle Scholar
Liu, J. and Xu, G. (2014). On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields. Ann. Appl. Prob. 24, 1691–1738.Google Scholar
Marcus, M. B. and Shepp, L. A. (1970). Continuity of Gaussian processes. Trans. Amer. Math. Soc. 151, 377–391.Google Scholar
Mitzenmacher, M. and Upfal, E. (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press.Google Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence, RI.Google Scholar
Sudakov, V. N. and Cirel'son, B. S. (1974). Extremal properties of half spaces for spherically invariant measures. Zap. Naučn. Sem. LOMI 41, 14–24.Google Scholar
Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Prob. 21, 34–71.Google Scholar
Talagrand, M. (1996). Majorizing measures: the generic chaining. Ann. Prob. 24, 1049–1103.Google Scholar
Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Prob. 33, 1362–1396.Google Scholar
Taylor, J. E. and Adler, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Prob. 31, 533–563.Google Scholar
Traub, J. F., Wasilokowski, G. W. and Woźniakowski, H. (1988). Information-Based Complexity. Academic Press, Boston, MA.Google Scholar
Tsirel'son, V. S. (1976). The density of the distribution of the maximum of a Gaussian process. Theory Prob. Appl. 20, 847–856.Google Scholar
Woźniakowski, H. (1997). Computational complexity of continuous problems. In Nonlinear Dynamics, Chaotic and Complex Systems, Cambridge University Press, pp. 283–295.Google Scholar