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Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Oleg Seleznjev
Oleg Seleznjev*
Affiliation:
Moscow State University
*
∗ Postal address: Faculty of Mathematics and Mechanics, Moscow State University, 119 899 Moscow, Russia.
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Abstract

We consider the piecewise linear interpolation of Gaussian processes with continuous sample paths and stationary increments. The interrelation between the smoothness of the incremental variance function, d(t – s) = E[(X(t) – X(s))2], and the interpolation errors in mean square and uniform metrics is studied. The method of investigation can also be applied to the analysis of different methods of interpolation. It is based on some limit results for large deviations of a sequence of Gaussian non-stationary processes and related point processes. Non-stationarity in our case means mainly the local stationary condition for the sequence of correlation functions rn(t,s), n = 1, 2, ···, which has to hold uniformly in n. Finally, we discuss some examples and an application to the calculation of the distribution function of the maximum of a continuous Gaussian process with a given precision.

Keywords

MAXIMUM OF NON-STATIONARY GAUSSIAN PROCESSESPOINT PROCESSINTERPOLATION OF REALIZATION

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G15: Gaussian processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 481 - 499
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Supported by the Swedish Natural Science Research Council, contract 927620 NFR 3741–307.

References

Adler, R. J. (1990) An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. (IMS Lecture Notes-Monograph Series 12). IMS, Hayward, CA.CrossRefGoogle Scholar
Belyaev, Yu. K. and Simonyan, A. H. (1979) Asymptotic properties of deviations of a sample path of a Gaussian process from approximation by broken line for decreasing width of quantization. In Random Processes and Fields. ed. Belyaev, Yu. K. pp 9–21. Moscow University Press, Moscow.Google Scholar
Berman, S. M. (1985) The maximum of a Gaussian process with nonconstant variance. Ann. Inst H. Poincaré Probab. Statist. 21, 383–391.Google Scholar
Berman, S. M. and Kono, N. (1989) The maximum of a Gaussian process with nonconstant variance: a sharp bound for the distribution tail. Ann. Prob. 17, 632–650.CrossRefGoogle Scholar
Borell, C. (1975) The Brunn-Minkowski inequality in Gauss space. Inventiones Math. 30, 207–215.CrossRefGoogle Scholar
Cramer, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Dobric, V., Marcus, M. B. and Weber, M. J. G. (1988) The distribution of large values of the supremum of a Gaussian process. Astérique 157–158, 95–127.Google Scholar
Dudley, R. M. (1973) Sample functions of the Gaussian process. Ann. Prob. 1, 66–103.CrossRefGoogle Scholar
Eplett, W. T. (1986) Approximation theory for simulation of continuous Gaussian processes. Prob. Theory Rel. Fields 73, 159–181.CrossRefGoogle Scholar
Fernique, X. (1975) Regularité des trajectoires des fonctions alétoires gaussienns. (Lecture Notes in Mathematics 480) pp. 1–96. Springer, Berlin.Google Scholar
Hasofer, A. M. (1987) Distribution of the maximum of a Gaussian process by a Monte-Carlo method. J. Sound Vibr. 112, 283–293.CrossRefGoogle Scholar
Hasofer, A. M. (1989) Continuous simulation of Gaussian processes with given spectrum. Proc. 5th Int. Conf on Struct. Safety and Reliability, San Francisco, Aug 7-11. ASCE, New York.Google Scholar
Konstant, D. G. and Piterbarg, V. I. (1993) Extreme values of the cyclostationary Gaussian random processes. J. Appl. Prob. 30, 82–97.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer, New York.CrossRefGoogle Scholar
Pickands, J. III (1969) Upcrossings probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–73.CrossRefGoogle Scholar
Piterbarg, V. I. (1988) Theory of Asymptotic Methods in Gaussian Random Processes and Fields. Moscow University Press, Moscow.Google Scholar
Piterbarg, V. I. and Prisyazhnyuk, V. P. (1979) Asymptotics of the probability of large excursions for a nonstationary Gaussian process. Theor. Prob. Math. Statist. 18, 131–143.Google Scholar
Pflug, G. (1982) A statistically important Gaussian process. Stoch. Proc. Appl. 13, 45–57.CrossRefGoogle Scholar
Powell, M. J. D. (1974) Approximation Theory and Methods. Cambridge University Press, Cambridge.Google Scholar
Rudzkis, R. O. (1985) On the probability of a large excursion of a nonstationary Gaussian process. Lithuanian Math. J. 25, 143–154.CrossRefGoogle Scholar
Seleznjev, O. V. (1989a) The best approximation of random processes and approximation of periodic random processes. Univ. Lund Statist. Res. Rep. 6, 1–32.Google Scholar
Seleznjev, O. V. (1989b) Estimate of tail probability for maximum of Gaussian field. Univ. Lund Statist. Res. Rep. 10, 1–10.Google Scholar
Seleznjev, O. V. (1991) Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes. J. Appl. Prob. 28, 17–32.CrossRefGoogle Scholar
Su, Y. and Cambanis, S. (1993) Sampling designs for estimation of a random process. Stoch. Proc. Appl. 46, 47–89.CrossRefGoogle Scholar
Talagrand, M. (1988) Small tails for the supremum of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 24, 307–315.Google Scholar
Ylvisaker, N. D. (1968) A note on the absence of tangencies in Gaussian sample paths. Ann. Math. Statist. 39, 261–262.CrossRefGoogle Scholar