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Asymptotic distribution of sum and maximum for Gaussian processes

  • Hwai-Chung Ho (a1) and William P. McCormick (a2)
Abstract

Let {X n , n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = E X 0 X n . Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

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Corresponding author
Postal address: Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, ROC.
∗∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA. Email address: bill@stat.uga.edu.
References
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Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks/Cole, Pacific Grove, CA.
Brockwell, P. J., and Davis, R. A. (1987). Time Series: Theory and Methods. Springer, New York.
Doukhan, P. (1994). Mixing, Properties and Examples (Lecture Notes in Statist. 85). Springer, New York.
Gray, A. (1990). Tubes. Addison-Wesley, Reading, MA.
Ho, H. C., and Hsing, T. (1996). On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. J. Appl. Prob. 33, 138145.
Hsing, T. (1995). A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables. Ann. Prob. 23, 938947.
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Stationary Sequences and Processes. Springer, New York.
McCormick, W. P. (1980). Weak convergence for the maxima of stationary Gaussian processes using random normalization. Ann. Prob. 8, 498510.
McCormick, W. P. (1999). A geometric approach to obtaining the distribution of the maximum for a class of random fields. Preprint.
Mittal, Y., and Ylvisaker, D. (1975). Limit distributions for the maxima of stationary Gaussian processes. Stoch. Proc. Appl. 3, 118.
Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Prob. 21, 3471.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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