IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v118y2008i9p1679-1705.html
   My bibliography  Save this article

On the ergodicity and mixing of max-stable processes

Author

Listed:
  • Stoev, Stilian A.

Abstract

Max-stable processes arise in the limit of component-wise maxima of independent processes, under appropriate centering and normalization. In this paper, we establish necessary and sufficient conditions for the ergodicity and mixing of stationary max-stable processes. We do so in terms of their spectral representations by using extremal integrals. The large classes of moving maxima and mixed moving maxima processes are shown to be mixing. Other examples of ergodic doubly stochastic processes and non-ergodic processes are also given. The ergodicity conditions involve a certain measure of dependence. We relate this measure of dependence to the one of Weintraub [K.S.Weintraub, Sample and ergodic properties of some min-stable processes, Ann. Probab. 19 (2) (1991) 706-723] and show that Weintraub's notion of '0-mixing' is equivalent to mixing. Consistent estimators for the dependence function of an ergodic max-stable process are introduced and illustrated over simulated data.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:9:p:1679-1705
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(07)00180-9
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Hardin, Clyde D., 1982. "On the spectral representation of symmetric stable processes," Journal of Multivariate Analysis, Elsevier, vol. 12(3), pages 385-401, September.
    2. Cambanis, Stamatis & Hardin, Clyde D. & Weron, Aleksander, 1987. "Ergodic properties of stationary stable processes," Stochastic Processes and their Applications, Elsevier, vol. 24(1), pages 1-18, February.
    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. 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.
    2. Hashorva, Enkelejd, 2018. "Representations of max-stable processes via exponential tilting," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2952-2978.
    3. Molchanov, Ilga & Schmutz, Michael & Stucki, Kaspar, 2012. "Invariance properties of random vectors and stochastic processes based on the zonoid concept," DES - Working Papers. Statistics and Econometrics. WS ws122014, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Richard A. Davis & Claudia Klüppelberg & Christina Steinkohl, 2013. "Statistical inference for max-stable processes in space and time," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(5), pages 791-819, November.
    5. 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.
    6. 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.
    7. Wang, Yizao, 2018. "Extremes of q-Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2979-3005.
    8. Engelke, Sebastian & Kabluchko, Zakhar, 2015. "Max-stable processes and stationary systems of Lévy particles," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4272-4299.
    9. 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.
    10. Kabluchko, Zakhar & Schlather, Martin, 2010. "Ergodic properties of max-infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 281-295, March.

    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. Wang, Yizao & Stoev, Stilian A. & Roy, Parthanil, 2012. "Decomposability for stable processes," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1093-1109.
    2. Arijit Chakrabarty & Parthanil Roy, 2013. "Group-Theoretic Dimension of Stationary Symmetric α-Stable Random Fields," Journal of Theoretical Probability, Springer, vol. 26(1), pages 240-258, March.
    3. Andreas Basse-O'Connor & Mark Podolskij, 2015. "On critical cases in limit theory for stationary increments Lévy driven moving averages," CREATES Research Papers 2015-57, Department of Economics and Business Economics, Aarhus University.
    4. Pipiras, Vladas, 2007. "Nonminimal sets, their projections and integral representations of stable processes," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1285-1302, September.
    5. Sławomir Kolodyński & Jan Rosiński, 2003. "Group Self-Similar Stable Processes in R d," Journal of Theoretical Probability, Springer, vol. 16(4), pages 855-876, October.
    6. Stilian Stoev & Murad S. Taqqu, 2005. "Asymptotic self‐similarity and wavelet estimation for long‐range dependent fractional autoregressive integrated moving average time series with stable innovations," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(2), pages 211-249, March.
    7. Hsing, Tailen, 1995. "Limit theorems for stable processes with application to spectral density estimation," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 39-71, May.
    8. Ibragimov, Ildar & Kabluchko, Zakhar & Lifshits, Mikhail, 2019. "Some extensions of linear approximation and prediction problems for stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2758-2782.
    9. Mazur, Stepan & Otryakhin, Dmitry & Podolskij, Mark, 2018. "Estimation of the linear fractional stable motion," Working Papers 2018:3, Örebro University, School of Business.
    10. Magdziarz, Marcin, 2009. "Correlation cascades, ergodic properties and long memory of infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3416-3434, October.
    11. Pérez-Abreu, Victor & Rocha-Arteaga, Alfonso, 1997. "On stable processes of bounded variation," Statistics & Probability Letters, Elsevier, vol. 33(1), pages 69-77, April.
    12. Rutkowski, Marek, 1995. "Left and right linear innovations for a multivariate S[alpha]S random variable," Statistics & Probability Letters, Elsevier, vol. 22(3), pages 175-184, February.
    13. Andreas Basse-O'Connor & Raphaël Lachièze-Rey & Mark Podolskij, 2015. "Limit theorems for stationary increments Lévy driven moving averages," CREATES Research Papers 2015-56, Department of Economics and Business Economics, Aarhus University.
    14. Mathias Mørck Ljungdahl & Mark Podolskij, 2022. "Multidimensional parameter estimation of heavy‐tailed moving averages," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 593-624, June.
    15. Vladas Pipiras & Murad S. Taqqu, 2004. "Dilated Fractional Stable Motions," Journal of Theoretical Probability, Springer, vol. 17(1), pages 51-84, January.
    16. Roman Ger & Michael Keane & Jolanta K. Misiewicz, 2000. "On Convolutions and Linear Combinations of Pseudo-Isotropic Distributions," Journal of Theoretical Probability, Springer, vol. 13(4), pages 977-995, October.
    17. Pipiras, Vladas & Taqqu, Murad S. & Abry, Patrice, 2003. "Can continuous-time stationary stable processes have discrete linear representations?," Statistics & Probability Letters, Elsevier, vol. 64(2), pages 147-157, August.
    18. Soltani, A.R. & Parvardeh, A., 2005. "Decomposition of discrete time periodically correlated and multivariate stationary symmetric stable processes," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1838-1859, November.
    19. Mathias Mørck Ljungdahl & Mark Podolskij, 2020. "A minimal contrast estimator for the linear fractional stable motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 381-413, July.
    20. Rybaczuk, M. & Weron, K., 1989. "Linearly coupled quantum oscillators with Lévy stable noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 160(3), pages 519-526.

    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:118:y:2008:i:9:p:1679-1705. 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.