The Annals of Applied Probability

On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields

Jingchen Liu and Gongjun Xu

Full-text: Open access


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 $\int_{T}e^{f(t)}\,dt$. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field $f$ (given that $\int_{T}e^{f(t)}\,dt$ exceeds a large value) in total variation. Based on this approximation, we show that the tail event of $\int_{T}e^{f(t)}\,dt$ is asymptotically equivalent to the tail event of $\sup_{T}\gamma(t)$ where $\gamma(t)$ 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 $P(\int_{T}e^{f(t)}\,dt>b)$ with a prescribed relative accuracy.

Article information

Ann. Appl. Probab. Volume 24, Number 4 (2014), 1691-1738.

First available in Project Euclid: 14 May 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 65C05: Monte Carlo methods

Gaussian process change of measure efficient simulation


Liu, Jingchen; Xu, Gongjun. On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields. Ann. Appl. Probab. 24 (2014), no. 4, 1691--1738. doi:10.1214/13-AAP960.

Export citation


  • [1] Aberg, S. and Guttorp, P. (2008). Distribution of the maximum in air pollution fields. Environmetrics 19 183–208.
  • [2] Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
  • [3] Adler, R. J., Blanchet, J. H. and Liu, J. (2012). Efficient Monte Carlo for high excursions of Gaussian random fields. Ann. Appl. Probab. 22 1167–1214.
  • [4] Adler, R. J., Samorodnitsky, G. and Taylor, J. E. (2013). High level excursion set geometry for non-Gaussian infinitely divisible random fields. Ann. Probab. 41 134–169.
  • [5] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • [6] Adler, R. J., Taylor, J. E. and Worsley, K. J. (2009). Applications of random fields and geometry: Foundations and case studies. Preprint.
  • [7] Ahsan, S. M. (1978). Portfolio selection in a lognormal securities market. Zeitschrift für Nationalokonomie—Journal of Economics 38 105–118.
  • [8] Asmussen, S. (2000). Ruin Probabilities. Advanced Series on Statistical Science & Applied Probability 2. World Scientific, River Edge, NJ.
  • [9] Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stochastic Process. Appl. 64 103–125.
  • [10] Asmussen, S. and Rojas-Nandayapa, L. (2008). Asymptotics of sums of lognormal random variables with Gaussian copula. Statist. Probab. Lett. 78 2709–2714.
  • [11] Azaïs, J.-M. and Wschebor, M. (2005). On the distribution of the maximum of a Gaussian field with $d$ parameters. Ann. Appl. Probab. 15 254–278.
  • [12] 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. Stochastic Process. Appl. 118 1190–1218.
  • [13] Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken, NJ.
  • [14] Basak, S. and Shapiro, A. (2001). Value-at-risk-based risk management: Optimal policies and asset prices. Review of Financial Studies 14 371–405.
  • [15] Berman, S. M. (1985). An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. J. Appl. Probab. 22 454–460.
  • [16] Blanchet, J. and Liu, J. (2010). Efficient importance sampling in ruin problems for multidimensional regularly varying random walks. J. Appl. Probab. 47 301–322.
  • [17] Blanchet, J. and Liu, J. (2012). Efficient simulation and conditional functional limit theorems for ruinous heavy-tailed random walks. Stochastic Process. Appl. 122 2994–3031.
  • [18] Blanchet, J. H. and Liu, J. (2008). State-dependent importance sampling for regularly varying random walks. Adv. in Appl. Probab. 40 1104–1128.
  • [19] Blanchet, J. H., Liu, J. and Yang, X. (2010). Monte Carlo for large credit portfolios with potentially high correlations. In Proceedings of the 2010 Winter Simulation Conference, Baltimore, MD.
  • [20] Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
  • [21] Campbell, M. J. (1994). Time-series regression for counts—An investigation into the relationship between sudden-infant-death-syndrome and environmental-temperature. J. Roy. Statist. Soc. Ser. A 157 191–208.
  • [22] Chan, K. S. and Ledolter, J. (1995). Monte Carlo EM estimation for time series models involving counts. J. Amer. Statist. Assoc. 90 242–252.
  • [23] 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). Lecture Notes in Math. 550 20–41. Springer, Berlin.
  • [24] Cox, D. R. (1955). Some statistical methods connected with series of events. J. R. Stat. Soc. Ser. B Stat. Methodol. 17 129–157; discussion, 157–164.
  • [25] Cox, D. R. and Isham, V. (1980). Point Processes. Chapman & Hall, London.
  • [26] Davis, R. A., Dunsmuir, W. T. M. and Wang, Y. (2000). On autocorrelation in a Poisson regression model. Biometrika 87 491–505.
  • [27] Deutsch, H. P. (2004). Derivatives and Internal Models, 3rd ed. Palgrave Macmillan, Basingstoke, UK.
  • [28] Duffie, D. and Pan, J. (1997). An overview of value at risk. The Journal of Derivatives 4 7–49.
  • [29] Dufresne, D. (2001). The integral of geometric Brownian motion. Adv. in Appl. Probab. 33 223–241.
  • [30] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
  • [31] Foss, S. and Richards, A. (2010). On sums of conditionally independent subexponential random variables. Math. Oper. Res. 35 102–119.
  • [32] Gadrich, T. and Adler, R. J. (1993). Slepian models for nonstationary Gaussian processes. J. Appl. Probab. 30 98–111.
  • [33] Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2000). Variance reduction techniques for estimating value-at-risk. Management Science 46 1349–1364.
  • [34] Grigoriu, M. (1989). Reliability of Daniels systems subject to quasistatic and dynamic non-stationary Gaussian load processes. Prob. Eng. Mech. 4 128–134.
  • [35] Hüsler, J. (1990). Extreme values and high boundary crossings of locally stationary Gaussian processes. Ann. Probab. 18 1141–1158.
  • [36] Hüsler, J., Piterbarg, V. and Zhang, Y. (2011). Extremes of Gaussian processes with random variance. Electron. J. Probab. 16 1254–1280.
  • [37] Landau, H. J. and Shepp, L. A. (1970). On the supremum of a Gaussian process. Sankhyā Ser. A 32 369–378.
  • [38] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • [39] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) 23. Springer, Berlin.
  • [40] Lindgren, G. (1970). Some properties of a normal process near a local maximum. Ann. Math. Statist. 41 1870–1883.
  • [41] Lindgren, G. (1979). Prediction of level crossings for normal processes containing deterministic components. Adv. in Appl. Probab. 11 93–117.
  • [42] Liu, J. (2012). Tail approximations of integrals of Gaussian random fields. Ann. Probab. 40 1069–1104.
  • [43] 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.
  • [44] Liu, J. and Xu, G. (2013). On the density functions of integrals of Gaussian random fields. Adv. in Appl. Probab. 45 398–424.
  • [45] Liu, J. and Xu, G. (2014). Supplement to “On the conditional distributions and the efficient simulations of exponential integrals of gaussian random fields.” DOI:10.1214/13-AAP960SUPP.
  • [46] Marcus, M. B. and Shepp, L. A. (1970). Continuity of Gaussian processes. Trans. Amer. Math. Soc. 151 377–391.
  • [47] Mitzenmacher, M. and Upfal, E. (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge Univ. Press, Cambridge.
  • [48] Nardi, Y., Siegmund, D. O. and Yakir, B. (2008). The distribution of maxima of approximately Gaussian random fields. Ann. Statist. 36 1375–1403.
  • [49] Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs 148. Amer. Math. Soc., Providence, RI.
  • [50] Sudakov, V. N. and Tsirelson, B. S. (1974). Extremal properties of half spaces for spherically invariant measures. Zap. Nauchn. Sem. LOMI 45 75–82.
  • [51] Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34–71.
  • [52] Talagrand, M. (1996). Majorizing measures: The generic chaining. Ann. Probab. 24 1049–1103.
  • [53] Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362–1396.
  • [54] Traub, J. F., Wasilkowski, G. W. and Woźniakowski, H. (1988). Information-based Complexity. Academic Press, Boston, MA.
  • [55] Woźniakowski, H. (1997). Computational complexity of continuous problems. In Nonlinear Dynamics, Chaotic and Complex Systems (Zakopane, 1995) 283–295. Cambridge Univ. Press, Cambridge.
  • [56] Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. in Appl. Probab. 24 509–531.
  • [57] Zeger, S. L. (1988). A regression model for time series of counts. Biometrika 75 621–629.

Supplemental materials

  • Supplementary material: Supplement to “On the conditional distributions and the efficient simulations of exponential integrals of gaussian random fields”. Proofs of Proposition 14 and Lemmas 17, 18, 20, 22, 23 and 24 are provided in the supplementary material.