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Efficient Simulation for the Maximum of Infinite Horizon Discrete-Time Gaussian Processes

Part of: Stochastic processes Probabilistic methods, simulation and stochastic differential equations Limit theorems

Published online by Cambridge University Press:  14 July 2016

Jose Blanchet  and
Chenxin Li
Jose Blanchet*
Affiliation:
Columbia University
Chenxin Li*
Affiliation:
Columbia University
*
∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 321 S. W. Mudd, 500 West 120th Street, New York, NY 10027, USA.
∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 321 S. W. Mudd, 500 West 120th Street, New York, NY 10027, USA.
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Abstract

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We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean and indexed by positive integers reaches a high level, say b. In great generality such a probability converges to 0 exponentially fast in a power of b. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called the target bridge sampler (TBS), which takes a polynomial (in b) number of function evaluations to achieve a small relative error. The procedure also yields samples of the underlying process conditioned on hitting b in finite time. In addition, we apply our method to the problem of estimating the tail of the maximum of a superposition of a large number, n, of independent Gaussian sources. In this situation TBS achieves a prescribed relative error with a bounded number of function evaluations as n ↗ ∞. A remarkable feature of TBS is that it is not based on exponential changes of measure. Our numerical experiments validate the performance indicated by our theoretical findings.

Keywords

Importance samplingrare-event simulationGaussian processlarge deviationsfractional Brownian noise

MSC classification

Secondary: 65C05: Monte Carlo methods 60G15: Gaussian processes 60F10: Large deviations
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 467 - 489
Copyright
Copyright © Applied Probability Trust 2011 

References

Addie, R., Mannersalo, P. and Norros, I. (1999). Most probable paths and performance formulae for buffers with Gaussian input traffic. Europ. Trans. Telecommun. 13, 183–196.CrossRefGoogle Scholar
Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
Adler, R. J., Blanchet, J. and Liu, J. (2008). Efficient simulation for tail probabilities of Gaussian random fields. In Proc. 40th Conf. Winter Simul., pp. 328–336.CrossRefGoogle Scholar
Adler, R. J., Blanchet, J. and Liu, J. (2010). Efficient Monte Carlo for large excursions of Gaussian random fields. Preprint. Available at http://arxiv.org/abs/1005.0812v3.Google Scholar
Alzer, H. (1997). On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389.Google Scholar
Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.CrossRefGoogle Scholar
Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole, Pacific Grove, CA.CrossRefGoogle Scholar
Blanchet, J., Glynn, P. and Lam, H. (2009). Rare event simulation for a slotted time M/G/s model. Queueing Systems 63, 33–57.CrossRefGoogle Scholar
Boots, N. K. and Mandjes, M. (2002). Fast simulation of a queue fed by a superposition of many (heavy-tailed) sources. Prob. Eng. Inf. Sci. 16, 205–232.Google Scholar
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Bucklew, J. A. and Radeke, R. (2003). On the Monte Carlo simulation of digital communication systems with Gaussian noise. IEEE Trans. Commun. 51, 267–274.CrossRefGoogle Scholar
Chang, J. T. and Peres, Y. (1997). Ladder heights, Gaussian random walks and the Riemann zeta function. Ann. Prob. 25, 787–802.Google Scholar
Cody, W. J. (1969). Rational Chebyshev approximations for the error function. Math. Comput. 23, 631–637.CrossRefGoogle Scholar
Debicki, K. (1999). A note on LDP for supremum of Gaussian processes over infinite horizon. Statist. Prob. Lett. 44, 211–219.CrossRefGoogle Scholar
Debicki, K. and Mandjes, M. (2003). Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Prob. 40, 704–720.CrossRefGoogle Scholar
Dieker, A. B. (2005). Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115, 207–248.Google Scholar
Dieker, A. B. and Mandjes, M. (2006). Fast simulation of overflow probabilities in a queue with Gaussian input. ACM Trans. Model. Comput. Simul. 16, 1–33.Google Scholar
Duffield, N. G. and O'Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Camb. Phil. Soc. 118, 363–374.Google Scholar
Dupuis, P. and Wang, H. (2004). Importance sampling, large deviations, and differential games. Stoch. Stoch. Reports 76, 481–508.Google Scholar
Durrett, R. (2004). Probability: Theory and Examples, 3rd edn. Duxbury Press, Belmont, CA.Google Scholar
Giordano, S., Gubinelli, M. and Pagano, M. (2007). Rare events of Gaussian processes: a performance comparison between bridge Monte-Carlo and importance sampling. In Next Generation Teletraffic and Wired/Wireless Advanced Networking (Lecture Notes Comput. Sci. 4712), Springer, Berlin, pp. 269–280.Google Scholar
Huang, C., Devetsikiotis, M., Lambadaris, I. and Kaye, A. R. (1999). Fast simulation of queues with long-range dependent traffic. Commun. Statist. Stoch. Models 15, 429–460.CrossRefGoogle Scholar
Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.Google Scholar
Juneja, S. and Shahabuddin, P. (2006). Rare event simulation techniques: an introduction and recent advances. In Handbook on Simulation, eds Henderson, S. and Nelson, B., North-Holland, Amsterdam, pp. 291–350.Google Scholar
Likhanov, N. and Mazumdar, R. R. (1999). Cell loss asymptotics in buffers fed with a large number of independent stationary sources. J. Appl. Prob. 36, 86–96.Google Scholar
Mandjes, M. (2007). Large Deviations for Gaussian Queues. John Wiley, Chichester.CrossRefGoogle Scholar
Michna, Z. (1999). On tail probabilities and first passage times for fractional Brownian motion. Math. Meth. Operat. Res. 49, 335–354.CrossRefGoogle Scholar
Norros, I. (1999). Busy periods of fractional Brownian storage: a large deviations approach. Adv. Performance Analysis 2, 1–19.Google Scholar
Pickands, J. III (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145, 75–86.Google Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence, RI.Google Scholar
Sadowsky, J. S. and Bucklew, J. A. (1990). On large deviations theory and asymptotically efficient Monte Carlo estimation. IEEE Trans. Inf. Theory 36, 579–588.CrossRefGoogle Scholar
Traub, J. F. (2003). Information-based complexity. In Encyclopedia of Computer Science, 4th edn. John Wiley, Chichester, pp. 850–854.Google Scholar