IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v65y1996i2p171-185.html
   My bibliography  Save this article

Computing the extremal index of special Markov chains and queues

Author

Listed:
  • Hooghiemstra, Gerard
  • Meester, Ludolf E.

Abstract

We consider extremal properties of Markov chains. Rootzén (1988) gives conditions for stationary, regenerative sequences so that the normalized process of level exceedances converges in distribution to a compound Poisson process. He also provides expressions for the extremal index and the compounding probabilities; in general it is not easy to evaluate these. We show how in a number of instances Markov chains can be coupled with two random walks which, in terms of extremal behaviour, bound the chain from above and below. Using a limiting argument it is shown that the lower bound converges to the upper one, yielding the extremal index and the compounding probabilities of the Markov chain. An FFT algorithm by Grübel (1991) for the stationary distribution of a G/G/1 queue is adapted for the extremal index; it yields approximate, but very accurate results. Compounding probabilities are calculated explicitly in a similar fashion. The technique is applied to the G/G/1 queue, G/M/c queues and ARCH processes, whose extremal behaviour de Haan et al. (1989) characterized using simulation.

Suggested Citation

  • Hooghiemstra, Gerard & Meester, Ludolf E., 1996. "Computing the extremal index of special Markov chains and queues," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 171-185, December.
    • Handle: RePEc:eee:spapps:v:65:y:1996:i:2:p:171-185
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(96)00111-1
    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. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    2. de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
    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. Evsey V. Morozov & Irina V. Peshkova & Alexander S. Rumyantsev, 2023. "Bounds and Maxima for the Workload in a Multiclass Orbit Queue," Mathematics, MDPI, vol. 11(3), pages 1-15, January.

    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. Danielsson, Jon & Jorgensen, Bjorn N. & Sarma, Mandira & de Vries, Casper G., 2006. "Comparing downside risk measures for heavy tailed distributions," Economics Letters, Elsevier, vol. 92(2), pages 202-208, August.
    2. Phornchanok Cumperayot & Casper G. de Vries, 2006. "Large Swings in Currencies driven by Fundamentals," Tinbergen Institute Discussion Papers 06-086/2, Tinbergen Institute.
    3. Danielsson, Jon & Ergun, Lerby M. & Haan, Laurens de & Vries, Casper G. de, 2016. "Tail index estimation: quantile driven threshold selection," LSE Research Online Documents on Economics 66193, London School of Economics and Political Science, LSE Library.
    4. Lux, Thomas & Alfarano, Simone, 2016. "Financial power laws: Empirical evidence, models, and mechanisms," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 3-18.
    5. Jalal, Amine & Rockinger, Michael, 2008. "Predicting tail-related risk measures: The consequences of using GARCH filters for non-GARCH data," Journal of Empirical Finance, Elsevier, vol. 15(5), pages 868-877, December.
    6. Borkovec, Milan, 2000. "Extremal behavior of the autoregressive process with ARCH(1) errors," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 189-207, February.
    7. Tian, Shuairu & Hamori, Shigeyuki, 2015. "Modeling interest rate volatility: A Realized GARCH approach," Journal of Banking & Finance, Elsevier, vol. 61(C), pages 158-171.
    8. Liu, Ming, 2000. "Modeling long memory in stock market volatility," Journal of Econometrics, Elsevier, vol. 99(1), pages 139-171, November.
    9. Hartmann, P. & Straetmans, S. & de Vries, C.G., 2010. "Heavy tails and currency crises," Journal of Empirical Finance, Elsevier, vol. 17(2), pages 241-254, March.
    10. Emeka Nkoro & Aham Kelvin Uko, 2016. "Exchange Rate and Inflation Volatility and Stock Prices Volatility: Evidence from Nigeria, 1986-2012," Journal of Applied Finance & Banking, SCIENPRESS Ltd, vol. 6(6), pages 1-4.
    11. Minot, Nicholas, 2014. "Food price volatility in sub-Saharan Africa: Has it really increased?," Food Policy, Elsevier, vol. 45(C), pages 45-56.
    12. Tomanova, Lucie, 2013. "Exchange Rate Volatility and the Foreign Trade in CEEC," EY International Congress on Economics I (EYC2013), October 24-25, 2013, Ankara, Turkey 267, Ekonomik Yaklasim Association.
    13. Bernard, Jean-Thomas & Idoudi, Nadhem & Khalaf, Lynda & Yelou, Clement, 2007. "Finite sample multivariate structural change tests with application to energy demand models," Journal of Econometrics, Elsevier, vol. 141(2), pages 1219-1244, December.
    14. Chang, Chia-Lin, 2015. "Modelling a latent daily Tourism Financial Conditions Index," International Review of Economics & Finance, Elsevier, vol. 40(C), pages 113-126.
    15. Hendry, David F. & Clements, Michael P., 2003. "Economic forecasting: some lessons from recent research," Economic Modelling, Elsevier, vol. 20(2), pages 301-329, March.
    16. Adugna Lemi & Sisay Asefa, 2009. "Differential Impacts of Economic Volatility and Governance on Manufacturing and Non-Manufacturing Foreign Direct Investments: The Case of US Multinationals in Africa," Eastern Economic Journal, Palgrave Macmillan;Eastern Economic Association, vol. 35(3), pages 367-395.
    17. Zia-Ur- Rahman, 2019. "Influence of Excessive Expenditure of the Government in Perspective of Interest Rate and Money Circulation Which in Turn Affects the Growing Process in Pakistan," Asian Journal of Economics and Empirical Research, Asian Online Journal Publishing Group, vol. 6(2), pages 120-129.
    18. Goncalves, Silvia & Kilian, Lutz, 2004. "Bootstrapping autoregressions with conditional heteroskedasticity of unknown form," Journal of Econometrics, Elsevier, vol. 123(1), pages 89-120, November.
    19. Li, Yuming, 1998. "Expected stock returns, risk premiums and volatilities of economic factors1," Journal of Empirical Finance, Elsevier, vol. 5(2), pages 69-97, June.
    20. Henry, Olan T. & Olekalns, Nilss & Suardi, Sandy, 2007. "Testing for rate dependence and asymmetry in inflation uncertainty: Evidence from the G7 economies," Economics Letters, Elsevier, vol. 94(3), pages 383-388, March.

    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:65:y:1996:i:2:p:171-185. 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.