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Probability tails of Gaussian extrema

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  • Samorodnitsky, Gennady

Abstract

We study the supremum of 'the' standard isonormal linear process L on a subset of a real Hilbert space H. Upper and lower bounds on the probability that supx[epsilon] LX>[lambda], [lambda] large, are found. We treat a number of examples. These include the distribution of the maximum of certain 'locally stationary' processes on 1, as well as those of the rectangle indexed, pinned Brownian sheet in k and the half-plane indexed pinned sheet in 2. We also consider Brownian motion indexed by convex sets in [0, 1]2.

Suggested Citation

  • Samorodnitsky, Gennady, 1991. "Probability tails of Gaussian extrema," Stochastic Processes and their Applications, Elsevier, vol. 38(1), pages 55-84, June.
    • Handle: RePEc:eee:spapps:v:38:y:1991:i:1:p:55-84
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    Citations

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    Cited by:

    1. Koning, A.J. & Protassov, V., 2001. "Tail behaviour of Gaussian processes with applications to the Brownian pillow," Econometric Institute Research Papers EI 2001-49, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Arvanitis, Stelios & Scaillet, Olivier & Topaloglou, Nikolas, 2020. "Spanning analysis of stock market anomalies under prospect stochastic dominance," Working Papers unige:134101, University of Geneva, Geneva School of Economics and Management.
    3. Albin, J. M. P., 1999. "Extremes of totally skewed [alpha]-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 185-212, February.
    4. Krzysztof Dȩbicki & Peng Liu & Zbigniew Michna, 2020. "Sojourn Times of Gaussian Processes with Trend," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2119-2166, December.
    5. Chen, Zhe & Leskelä, Lasse & Viitasaari, Lauri, 2019. "Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2723-2757.
    6. Hjort, N.L. & Koning, A.J., 2001. "Constancy of distributions: nonparametric monitoring of probability distributions over time," Econometric Institute Research Papers EI 2001-50, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    7. Albin, J.M.P., 2018. "On covariance functions with slowly or regularly varying modulo of continuity," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 177-182.
    8. Koning, Alex J. & Protasov, Vladimir, 2003. "Tail behaviour of Gaussian processes with applications to the Brownian pillow," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 370-397, November.
    9. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Rolski, Tomasz, 2018. "Extremal behavior of hitting a cone by correlated Brownian motion with drift," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4171-4206.

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