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Extremes of multidimensional Gaussian processes

Author

Listed:
  • Debicki, K.
  • Kosinski, K.M.
  • Mandjes, M.
  • Rolski, T.

Abstract

This paper considers extreme values attained by a centered, multidimensional Gaussian process X(t)=(X1(t),...,Xn(t)) minus drift d(t)=(d1(t),...,dn(t)), on an arbitrary set T. Under mild regularity conditions, we establish the asymptotics of for positive thresholds qi>0, i=1,...,n and u-->[infinity]. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases. A number of examples illustrate the theory.

Suggested Citation

  • Debicki, K. & Kosinski, K.M. & Mandjes, M. & Rolski, T., 2010. "Extremes of multidimensional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2289-2301, December.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:12:p:2289-2301
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    References listed on IDEAS

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    1. Hüsler, J. & Piterbarg, V., 1999. "Extremes of a certain class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 257-271, October.
    2. Avram, Florin & Palmowski, Zbigniew & Pistorius, Martijn, 2008. "A two-dimensional ruin problem on the positive quadrant," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 227-234, February.
    3. Dieker, A.B., 2005. "Extremes of Gaussian processes over an infinite horizon," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 207-248, February.
    4. Dieker, A.B., 2005. "Conditional limit theorems for queues with Gaussian input, a weak convergence approach," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 849-873, May.
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    1. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2015. "Extremes of vector-valued Gaussian processes: Exact asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4039-4065.
    2. Krystecki, Konrad, 2022. "Parisian ruin probability for two-dimensional Brownian risk model," Statistics & Probability Letters, Elsevier, vol. 182(C).
    3. Remco Hofstad & Harsha Honnappa, 2019. "Large deviations of bivariate Gaussian extrema," Queueing Systems: Theory and Applications, Springer, vol. 93(3), pages 333-349, December.
    4. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Wang, Longmin, 2020. "Extremes of vector-valued Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5802-5837.
    5. Ji, Lanpeng & Peng, Xiaofan, 2023. "Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 418-452.
    6. 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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