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The Dependence Structure for Symmetric \(\alpha \)-stable CARMA(p,q) Processes

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Cyclostationarity: Theory and Methods - II (CSTA 2014)

Part of the book series: Applied Condition Monitoring ((ACM,volume 3))

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Abstract

In this paper, we investigate the dependence structure for the symmetric \(\alpha \)-stable CARMA(p,q) processes (i.e. continuous ARMA(p,q) models with symmetric \(\alpha \)-stable L\(\acute{\mathrm{e}}\)vy motion), that are a natural extension of second-order L\(\acute{\mathrm{e}}\)vy-driven CARMA processes. They are also the extension of discrete ARMA models with symmetric \(\alpha \)-stable innovations. For the considered stable models, the covariance function is not defined and therefore other measures of dependence have to be used. After determining the form of solution of considered continuous-time models, we study the codifference and the covariation—the most popular measures of dependence defined for symmetric \(\alpha \)-stable random variables. We prove the codifference and covariation are asymptotically proportional with the coefficient of proportionality equal to \(\alpha \). The result is similar to that obtained for discrete time series models. We also consider the alternative measure defined for infinitely divisible stochastic processes called the L\(\acute{\mathrm{e}}\)vy correlation cascade. As a special case, we consider symmetric \(\alpha \)-stable CAR(1) process, also called the Ornstein-Uhlenbeck process. In order to illustrate theoretical results, we analyze the real financial data that we model by using the examined processes.

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References

  1. Barndorff-Nielsen, O., & Shepardt, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society, Series B, 63, 1–42.

    Google Scholar 

  2. Brockwell, P. J. (2001). L\(\acute{{\rm e}}\)vy-driven CARMA processes. Annals of the Institute of Statistical Mathematics, 53(1), 113–124.

    Google Scholar 

  3. Brockwell, P. J., & Marquardt, T. (2005). L\(\acute{{\rm e}}\)vy-driven and fractionally integrated ARMA processes with continuous time parameter. Statistica Sinica, 15, 477–494.

    Google Scholar 

  4. Broszkiewicz-Suwaj, E., Makagon, A., Weron, R., & Wyłomańska Agnieszka, A. (2004). On detecting and modeling periodic correlation in financial data. Physica A, 336, 196–205.

    Google Scholar 

  5. Burnecki, K., Wyłomańska, A., Beletskii, A., Gonchar, V., & Chechkin, A. (2012). Recognition of stable distribution with Levy index alpha close to 2. Physical Review E, 85, 056711.

    Google Scholar 

  6. Eliazar, I., & Klafter, J. (2007). Correlation cascades of L\(\acute{{\rm e}}\)vy-driven random processes. Physica A, 376, 1–26.

    Google Scholar 

  7. Gajda, J., & Wyłomańska, A. (2012). Geometric Brownian motion with tempered stable waiting times. Journal of Statistical Physics, 148, 296–305.

    Google Scholar 

  8. Jarque, C. M., & Bera, A. K. (1980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters, 6(3), 255–259.

    Google Scholar 

  9. Janczura, J., Orzeł, S., & Wyłomańska, A. (2011). Subordinated alpha-stable Ornstein-Uhlenbeck process as a tool of financial data description. Physica A, 390, 4379–4387.

    Google Scholar 

  10. Janicki, A., & Weron, A. (1994). Can one see \(\alpha \)-stable variables and processes? Statistical Science, 9, 109–126.

    Google Scholar 

  11. Janicki, A., & Weron, A. (1994). Simulation and chaotic behaviour of \(\alpha \)-stable stochastic processes. New York: Marcel Dekker.

    Google Scholar 

  12. MacCulloch, J. H. (1986). Simple consistent estimators of stable distribution parameters. Communications in Statistics-Simulation and Computation, 15, 1109–1136.

    Google Scholar 

  13. Maciejewska, M., Szczurek, A., Janczura, J., & Wyłomańska, A. (2013). Stochastic modeling of indoor air temperature. Journal of Statistical Physics, 152, 979–994.

    Google Scholar 

  14. Magdziarz, M. (2009). Correlation cascades, Ergodic properties and long memory of infinitely divisible processes. Stochastic Processes and Their Applications, 119, 3416–3434.

    Google Scholar 

  15. Marquardt, T. (2006). Fractional L\(\acute{{\rm e}}\)vy processes, CARMA processes and related topics. Doctoral Thesis, TU Munchen.

    Google Scholar 

  16. Marquardt, T., & Stelzer, R. (2007). Multivariate CARMA processes. Stochastic Processes and Their Applications, 117, 96–120.

    Google Scholar 

  17. Mikosch, T., Gadrich, T., Klueppelberg, C., & Adler, R. J. (1995). Parameter estimation form ARMA models with infinite variance innovations. Annals of Statistics, 23(1), 305–326.

    Google Scholar 

  18. Mittnik, S., & Rachev, S. T. (2000). Stable paretian models in finance. New York: Wiley.

    Google Scholar 

  19. Nowicka, J. (1997). Asymptotic behavior of the covariation and the codifference for ARMA models with stable innovations. Stochastic Models, 13, 673–685.

    Google Scholar 

  20. Nowicka-Zagrajek, J., & Wyłomańska, A. (2006). The dependence structure for PARMA models with \(\alpha -\)stable innovations. Acta Physica Polonica, 37(11), 3071–3082.

    Google Scholar 

  21. Nowicka-Zagrajek, J., & Wyłomańska, A. (2008). Measures of dependence for stable AR(1) models with time-varying coefficients. Stochastic Models, 24(1), 58–70.

    Google Scholar 

  22. Rosadi, D. (2005). Asymptotic behavior of the codifference and the dynamical function for ARMA models with infinite variance. Journal of Indonesian Mathematical Society (MIHMI), 11(1), 59–69.

    Google Scholar 

  23. Rosadi, D., & Deistler, M. (2011). Estimating the codifference function of linear time series models with infinite variance. Metrika, 73(3), 395–429.

    Google Scholar 

  24. Obuchowski, J., & Wyłomańska, A. (2013). The Ornstein-Uhlenbeck process with non-Gaussian structure. Acta Physica Polonica A, 44(5), 1123–1136.

    Google Scholar 

  25. Obuchowski, J., Wyłomańska, A., & Zimroz, R. (2014). Selection of informative frequency band in local damage detection in rotating machinery. Mechanical Systems and Signal Processing, 48, 138–152.

    Google Scholar 

  26. Samorodnitsky, G., & Taqqu, M. S. (1994). Stable non-Gaussian random processes. New York: Chapman & Hall.

    Google Scholar 

  27. Stuck, B. W., & Kleiner, B. (1974). A statistical analysis of telephone noise. Bell System Technical Journal, 53, 1263–1320.

    Google Scholar 

  28. Uchaikin, V. V., & Zolotarev, V. M. (1999). Chance and stability, stable distributions and their applications. Utrecht: VSP.

    Google Scholar 

  29. Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the theory of the Brownian motion. Physical Review, 36, 823–841.

    Google Scholar 

  30. Vasi\(\check{\text{ c }}\)ek, O. (1977). An equilibrium characterisation of the term structure. Journal of Financial Economics, 5(2), 177–188.

    Google Scholar 

  31. Wyłomańska, A. (2011). Measures of dependence for Ornstein-Uhlenbeck process with tempered stable distribution. Acta Physica Polonica, 42(10), 2049–2062.

    Google Scholar 

  32. Wyłomańska, A., Chechkin, A., Sokolov, I., & Gajda, J. (2015). Codifference as a practical tool to measure interdependence. Physica A, 421, 412–429.

    Google Scholar 

  33. Wyłomańska, A., Obuchowski, J., Zimroz, R., Hurd, H. (2014). Periodic autoregressive modeling of vibration time series from planetary gearbox used in bucket wheel excavator, In Fakher Chaari et al. (Eds.), Cyclostationarity: Theory and methods. Lecture Notes in Mechanical Engineering, (pp. 171–186).

    Google Scholar 

  34. Yu, G., & Li, Ch. (2013). A new statistical modeling and detection method for rolling element bearings faults based on alpha-stable distribution. Mechanical Systems and Signal Processing, 41, 155–175.

    Google Scholar 

  35. Zak, G., Obuchowski, J., Wyłomańska, A., & Zimroz, R. (2014). Application of ARMA modelling and alpha-stable distribution for local damage detection in bearings. Diagnostyka, 15(3), 3–11.

    Google Scholar 

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Authors and Affiliations

  1. Hugo Steinhaus Center, Department of Mathematics, Wrocław University of Technology, Wrocław, Poland

    Agnieszka Wylomanska

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  1. Agnieszka Wylomanska
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Correspondence to Agnieszka Wylomanska .

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Editors and Affiliations

  1. National School of Engineers of Sfa, Sfax, Tunisia

    Fakher Chaari

  2. Institute of Mathematics, Cracow University of Technology, Krakow, Poland

    Jacek Leskow

  3. Department of Engineering, University of Napoli “Parthenope”, Napoli, Italy

    Antonio Napolitano

  4. Institute of Mining Engineering, Wroclaw University of Technology, Wroclaw, Poland

    Radoslaw Zimroz

  5. Institute of Mathematics and Computer Sc, Wroclaw University of Technology, Wroclaw, Poland

    Agnieszka Wylomanska

  6. Department of Applied Mathematics, University of Science and Technology, Krakow, Poland

    Anna Dudek

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Wylomanska, A. (2015). The Dependence Structure for Symmetric \(\alpha \)-stable CARMA(p,q) Processes. In: Chaari, F., Leskow, J., Napolitano, A., Zimroz, R., Wylomanska, A., Dudek, A. (eds) Cyclostationarity: Theory and Methods - II. CSTA 2014. Applied Condition Monitoring, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-16330-7_10

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  • DOI: https://doi.org/10.1007/978-3-319-16330-7_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-16329-1

  • Online ISBN: 978-3-319-16330-7

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