The Annals of Applied Probability

On convergence of the uniform norms for Gaussian processes and linear approximation problems

J. Hüsler, V. Piterbarg, and O. Seleznjev

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Abstract

We consider the large values and the mean of the uniform norms for a sequence of Gaussian processes with continuous sample paths. The convergence of the normalized uniform norm to the standard Gumbel (or double exponential) law is derived for distributions and means. The results are obtained from the Poisson convergence of the associated point process of exceedances for a general class of Gaussian processes. As an application we study the piecewise linear interpolation of Gaussian processes whose local behavior is like fractional (integrated fractional) Brownian motion (or with locally stationary increments). The overall interpolation performance for the random process is measured by the $p$th moment of the approximation error in the uniform norm. The problem of constructing the optimal sets of observation locations (or interpolation knots) is done asymptotically, namely, when the number of observations tends to infinity. The developed limit technique for a sequence of Gaussian nonstationary processes can be applied to analysis of various linear approximation methods.

Article information

Source
Ann. Appl. Probab., Volume 13, Number 4 (2003), 1615-1653.

Dates
First available in Project Euclid: 25 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1069786514

Digital Object Identifier
doi:10.1214/aoap/1069786514

Mathematical Reviews number (MathSciNet)
MR2023892

Zentralblatt MATH identifier
1038.60040

Subjects
Primary: 60G70: Extreme value theory; extremal processes 60G15: Gaussian processes 60F05: Central limit and other weak theorems

Keywords
Maxima of Gaussian processes uniform norm $p$th moment convergence piecewise linear approximation fractional Brownian motion

Citation

Hüsler, J.; Piterbarg, V.; Seleznjev, O. On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13 (2003), no. 4, 1615--1653. doi:10.1214/aoap/1069786514. https://projecteuclid.org/euclid.aoap/1069786514


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References

  • Belyaev, Yu. K. and Simonyan, A. H. (1979). Asymptotic properties of deviations of a sample path of a Gaussian process from approximation by regression broken line for decreasing width of quantization. In Random Processes and Fields (Yu. K. Belyaev, ed.) 9--21. Moscow Univ. Press.
  • Berman, S. M. (1971). Maxima and high level excursions of stationary Gaussian processes. Trans. Amer. Math. Soc. 160 65--85.
  • Berman, S. M. (1974). Sojourns and extremes of Gaussian process. Ann. Probab. 2 999--1026. [Corrections 8 (1980) 999; 12 (1984) 281.]
  • Berman, S. M. (1985). The maximum of a Gaussian process with nonconstant variance. Ann. Inst. H. Poincaré Probab. Statist. 21 383--391.
  • Bräcker, H. U. (1993). High boundary excursions of locally stationary Gaussian processes. Ph.D. thesis, Univ. Bern.
  • Buslaev, A. P. and Seleznjev, O. (1999). On certain extremal problems in approximation theory of random processes. East J. Approx. 5 467--481.
  • Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. Wiley, New York.
  • Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.
  • Eplett, W. T. (1986). Approximation theory for simulation of continuous Gaussian processes. Probab. Theory Related Fields 73 159--181.
  • Hüsler, J. (1983). Asymptotic approximation of crossing probabilities of random sequences. Z. Wahrsch. Verw. Gebiete 63 257--270.
  • Hüsler, J. (1990). Extreme values and high boundary crossings for locally stationary Gaussian processes. Ann. Probab. 18 1141--1158.
  • Hüsler, J. (1995). A note on extreme values of locally stationary Gaussian processes. J. Statist. Plann. Inference 45 203--213.
  • Hüsler, J. (1999). Extremes of Gaussian processes, on results of Piterbarg and Seleznjev. Statist. Probab. Lett. 44 251--258.
  • Johnson, N. L. and Kotz, S. (1970). Distributions in Statistics: Continuous Univariate Distributions 1. Wiley, New York.
  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Lindgren, G., de Maré, J. and Rootzén, H. (1975). Weak convergence of high level crossings and maxima for one or more Gaussian processes. Ann. Probab. 3 961--978.
  • Müller-Gronbach, T. (1996). Optimal designs for approximating the path of a stochastic process. J. Statist. Plann. Inference 49 371--385.
  • Müller-Gronbach, T. and Ritter, K. (1997). Uniform reconstruction of Gaussian processes. Stochastic Process. Appl. 69 55--70.
  • Pickands, J., III. (1968). Moment convergence of sample extremes. Ann. Math. Statist. 39 881--889.
  • Pickands, J., III. (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51--73.
  • Piterbarg, V. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Amer. Math. Soc., Providence, RI.
  • Piterbarg, V. I. and Prisyazhn'uk, V. (1978). Asymptotic behaviour of the probability of a large excursion of a non-stationary Gaussian process. Theory Probab. Math. Statist. 18 121--133.
  • Piterbarg, V. and Seleznjev, O. (1994). Linear interpolation of random processes and extremes of a sequence of Gaussian non-stationary processes. Technical Report 1994:446, Center Stoch. Process, North Carolina Univ., Chapel Hill.
  • Qualls, C. and Watanabe, H. (1972). Asymptotic properties of Gaussian processes. Ann. Math. Statist. 43 580--596.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Ritter, K. (1999). Average Case Analysis of Numerical Problems. Lecture Notes in Math. 1733. Springer, New York.
  • Ritter, K., Wasilkowski, G. W. and Woźniakowski, W. (1995). Multivariate integration and approximation for random fields satisfying Sacks--Ylvisaker conditions. Ann. Appl. Probab. 5 518--540.
  • Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiment. Statist. Sci. 4 409--435.
  • Sacks, J. and Ylvisaker, D. (1966). Design for regression problems with correlated errors. Ann. Math. Statist. 37 66--89.
  • Seleznjev, O. (1989). The best approximation of random processes and approximation of periodic random processes. Research Report 1989:6, Dept. Mathematical Statistics, Lund University, Sweden.
  • Seleznjev, O. (1991). Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes. J. Appl. Probab. 28 17--32.
  • Seleznjev, O. (1993). Limit theorems for maxima and crossings of sequence of nonstationary Gaussian processes and interpolation of random processes. Report 1993:8, Dept. Mathematical Statistics, Lund University, Sweden.
  • Seleznjev, O. (1996). Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments. Adv. in Appl. Probab. 28 481--499.
  • Seleznjev, O. (1999). Linear approximation of random processes and sampling design problems. In Probability Theory and Mathematical Statistics (B. Grigelionis, J. Kubilius, V. Paulauskas, H. Pragauskas and V. Statulevicius, eds.) 665--684. VSP/TEV, The Netherlands.
  • Seleznjev, O. (2000). Spline approximation of random processes and design problems. J. Statist. Plann. Inference 84 249--262.
  • Su, Y. and Cambanis, S. (1993). Sampling designs for estimation of a random process. Stochastic Process. Appl. 46 47--89.
  • Wahba, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia.
  • Weba, M. (1992). Simulation and approximation of stochastic processes by spline functions. SIAM J. Sci. Comput. 13 1085--1096.