IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v128y2018i9p2952-2978.html
   My bibliography  Save this article

Representations of max-stable processes via exponential tilting

Author

Listed:
  • Hashorva, Enkelejd

Abstract

The recent contribution (Dieker and Mikosch, 2015) obtained representations of max-stable stationary Brown–Resnick process ζZ(t),t∈Rd with spectral process Z being Gaussian. With motivations from Dieker and Mikosch (2015) we derive for general Z, representations for ζZ via exponential tilting of Z. Our findings concern Dieker–Mikosch representations of max-stable processes, two-sided extensions of stationary max-stable processes, inf-argmax representation of max-stable distributions, and new formulas for generalised Pickands constants. Our applications include conditions for the stationarity of ζZ, a characterisation of Gaussian distributions and an alternative proof of Kabluchko’s characterisation of Gaussian processes with stationary increments.

Suggested Citation

  • Hashorva, Enkelejd, 2018. "Representations of max-stable processes via exponential tilting," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2952-2978.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:9:p:2952-2978
    DOI: 10.1016/j.spa.2017.10.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414917302545
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2017.10.003?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Falk, Michael, 2015. "On idempotent D-norms," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 283-294.
    2. Davis, Richard A. & Mikosch, Thomas & Zhao, Yuwei, 2013. "Measures of serial extremal dependence and their estimation," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2575-2602.
    3. Segers, Johan & Zhao, Yuwei & Meinguet, Thomas, 2017. "Polar decomposition of regularly varying time series in star-shaped metric spaces," LIDAM Reprints ISBA 2017029, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Padoan, Simone A., 2011. "Multivariate extreme models based on underlying skew-t and skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 977-991, May.
    5. Molchanov, Ilya & Strokorb, Kirstin, 2016. "Max-stable random sup-measures with comonotonic tail dependence," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2835-2859.
    6. Molchanov, Ilya & Stucki, Kaspar, 2013. "Stationarity of multivariate particle systems," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2272-2285.
    7. Robert, Christian Y., 2013. "Some new classes of stationary max-stable random fields," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1496-1503.
    8. Opitz, T., 2013. "Extremal t processes: Elliptical domain of attraction and a spectral representation," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 409-413.
    9. Kabluchko, Zakhar & Wang, Yizao, 2014. "Limiting distribution for the maximal standardized increment of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2824-2867.
    10. Dombry, Clément & Kabluchko, Zakhar, 2017. "Ergodic decompositions of stationary max-stable processes in terms of their spectral functions," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1763-1784.
    11. Falk, Michael & Stupfler, Gilles, 2017. "An offspring of multivariate extreme value theory: The max-characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 85-95.
    12. Stoev, Stilian A., 2008. "On the ergodicity and mixing of max-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1679-1705, September.
    13. C. Dombry & F. Éyi-Minko & M. Ribatet, 2013. "Conditional simulation of max-stable processes," Biometrika, Biometrika Trust, vol. 100(1), pages 111-124.
    14. Hashorva, Enkelejd, 2007. "Extremes of conditioned elliptical random vectors," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1583-1591, September.
    15. Kume, Alfred & Hashorva, Enkelejd, 2012. "Calculation of Bayes premium for conditional elliptical risks," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 632-635.
    16. Marc G. Genton & Simone A. Padoan & Huiyan Sang, 2015. "Multivariate max-stable spatial processes," Biometrika, Biometrika Trust, vol. 102(1), pages 215-230.
    17. Clément Dombry & Sebastian Engelke & Marco Oesting, 2016. "Exact simulation of max-stable processes," Biometrika, Biometrika Trust, vol. 103(2), pages 303-317.
    18. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
    19. Rootzen, Holger & Segers, Johan & Wadsworth, Jenny, 2016. "Multivariate peaks over thresholds models," LIDAM Discussion Papers ISBA 2016018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    20. Wang, Yizao & Stoev, Stilian A., 2010. "On the association of sum- and max-stable processes," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 480-488, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Krzysztof Dȩbicki & Enkelejd Hashorva, 2020. "Approximation of Supremum of Max-Stable Stationary Processes & Pickands Constants," Journal of Theoretical Probability, Springer, vol. 33(1), pages 444-464, March.
    2. Bai, Long, 2020. "Extremes of standard multifractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 159(C).
    3. Krzysztof Dȩbicki & Zbigniew Michna & Xiaofan Peng, 2019. "Approximation of Sojourn Times of Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1183-1213, December.
    4. Hashorva, Enkelejd & Kume, Alfred, 2021. "Multivariate max-stable processes and homogeneous functionals," Statistics & Probability Letters, Elsevier, vol. 173(C).
    5. E. Hashorva, 2018. "Approximation of Some Multivariate Risk Measures for Gaussian Risks," Papers 1803.06922, arXiv.org, revised Oct 2018.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hashorva, Enkelejd & Kume, Alfred, 2021. "Multivariate max-stable processes and homogeneous functionals," Statistics & Probability Letters, Elsevier, vol. 173(C).
    2. Krzysztof Dȩbicki & Enkelejd Hashorva, 2020. "Approximation of Supremum of Max-Stable Stationary Processes & Pickands Constants," Journal of Theoretical Probability, Springer, vol. 33(1), pages 444-464, March.
    3. Janßen, Anja, 2019. "Spectral tail processes and max-stable approximations of multivariate regularly varying time series," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1993-2009.
    4. Koch, Erwan & Robert, Christian Y., 2022. "Stochastic derivative estimation for max-stable random fields," European Journal of Operational Research, Elsevier, vol. 302(2), pages 575-588.
    5. Hentschel, Manuel & Engelke, Sebastian & Segers, Johan, 2022. "Statistical Inference for Hüsler–Reiss Graphical Models Through Matrix Completions," LIDAM Discussion Papers ISBA 2022032, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Zhong, Peng & Huser, Raphaël & Opitz, Thomas, 2024. "Exact Simulation of Max-Infinitely Divisible Processes," Econometrics and Statistics, Elsevier, vol. 30(C), pages 96-109.
    7. Belzile, Léo R. & Nešlehová, Johanna G., 2017. "Extremal attractors of Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 68-92.
    8. Lee, Xing Ju & Hainy, Markus & McKeone, James P. & Drovandi, Christopher C. & Pettitt, Anthony N., 2018. "ABC model selection for spatial extremes models applied to South Australian maximum temperature data," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 128-144.
    9. Hofert, Marius & Huser, Raphaël & Prasad, Avinash, 2018. "Hierarchical Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 195-211.
    10. Kokoszka, Piotr & Kulik, Rafał, 2023. "Principal component analysis of infinite variance functional data," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    11. Dombry, Clément & Kabluchko, Zakhar, 2017. "Ergodic decompositions of stationary max-stable processes in terms of their spectral functions," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1763-1784.
    12. Buriticá, Gloria & Mikosch, Thomas & Wintenberger, Olivier, 2023. "Large deviations of ℓp-blocks of regularly varying time series and applications to cluster inference," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 68-101.
    13. Davis, Richard A. & Mikosch, Thomas & Zhao, Yuwei, 2013. "Measures of serial extremal dependence and their estimation," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2575-2602.
    14. Durieu, Olivier & Wang, Yizao, 2022. "Phase transition for extremes of a stochastic model with long-range dependence and multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 55-88.
    15. Segers, Johan, 2019. "One- versus multi-component regular variation and extremes of Markov trees," LIDAM Discussion Papers ISBA 2019001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Parthanil Roy, 2017. "Maxima of stable random fields, nonsingular actions and finitely generated abelian groups: A survey," Indian Journal of Pure and Applied Mathematics, Springer, vol. 48(4), pages 513-540, December.
    17. Clémençon, Stephan & Huet, Nathan & Sabourin, Anne, 2024. "Regular variation in Hilbert spaces and principal component analysis for functional extremes," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    18. Pedersen, Rasmus Søndergaard, 2016. "Targeting Estimation Of Ccc-Garch Models With Infinite Fourth Moments," Econometric Theory, Cambridge University Press, vol. 32(2), pages 498-531, April.
    19. R de Fondeville & A C Davison, 2018. "High-dimensional peaks-over-threshold inference," Biometrika, Biometrika Trust, vol. 105(3), pages 575-592.
    20. Moreno Bevilacqua & Alfredo Alegria & Daira Velandia & Emilio Porcu, 2016. "Composite Likelihood Inference for Multivariate Gaussian Random Fields," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(3), pages 448-469, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:128:y:2018:i:9:p:2952-2978. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.