Electronic Journal of Statistics

Weak dependence and GMM estimation of supOU and mixed moving average processes

Imma Valentina Curato and Robert Stelzer

Full-text: Open access


We consider a mixed moving average (MMA) process $X$ driven by a Lévy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Lévy basis. Using this property, we show conditions ensuring that sample mean and autocovariances of $X$ have a limiting normal distribution. We extend these results to stochastic volatility models and then investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter. For these estimators, we analyze the asymptotic behavior in detail.

Article information

Electron. J. Statist., Volume 13, Number 1 (2019), 310-360.

Received: July 2018
First available in Project Euclid: 9 February 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions 60G10: Stationary processes 60G51: Processes with independent increments; Lévy processes 60G57: Random measures
Secondary: 62F12: Asymptotic properties of estimators

Weak dependence Lévy basis generalized method of moments Ornstein-Uhlenbeck type process stochastic volatility

Creative Commons Attribution 4.0 International License.


Curato, Imma Valentina; Stelzer, Robert. Weak dependence and GMM estimation of supOU and mixed moving average processes. Electron. J. Statist. 13 (2019), no. 1, 310--360. doi:10.1214/18-EJS1523. https://projecteuclid.org/euclid.ejs/1549681240

Export citation


  • [1] Andrews, D. W. K. (1984). Non-strong mixing autoregressive processes., J. Appl. Probab. 21 930–934.
  • [2] Bardet, J. M., Doukhan, P. and León, J. R. (2008). A uniform central limit theorem for the periodogram and its applications to Whittle parametric estimation for weakly dependent time series., J. Time Series Anal. 29 906–945.
  • [3] Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein-Uhlenbeck type processes., Theory Probab. Appl. 45 175–194.
  • [4] Barndorff-Nielsen, O. E, Benth, F. and Veraart, E. D. (2015). Cross-commodity modelling by multivariate ambit fields, in, Commodities, Energy and Environmental Finance, edited by R. Aïd, M. Ludkovski and R. Sircar. Springer, New York.
  • [5] Barndorff-Nielsen, O. E, Hedevang, E., Schmiegel, J. and Szozda, B. (2016). Some recent developments in ambit stochastics, in, Stochastics of Environmental and Financial Economics, edited by F. Benth and G. Di Nunno. Springer, Cham.
  • [6] Barndorff-Nielsen, O. E., Nicolato, E. and Shepard, N. (2002). Some recent developments in stochastic volatility modeling. Special issue on volatility modelling., Quant. Finance 2 11–23.
  • [7] Barndorff-Nielsen, O. E. and Leonenko, N. N. (2005). Spectral properties of superpositions of Ornstein-Uhlenbeck type processes, methodology and computing., Methodol. Comput. Appl. Probab. 7 335–352.
  • [8] Barndorff-Nielsen, O. E. and Schmiegel, J. (2007). Ambit processes: with applications to turbulence and cancer growth, in, Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symposia, vol. 2, edited by F.E. Benth, G. Di Nunno, T. Lindstrom, B. Øksendal and T. Zhang. Springer, Berlin.
  • [9] Barndorff-Nielsen, O. E. and Shepard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics., J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241.
  • [10] Barndorff-Nielsen, O. E. and Stelzer, R. (2011). Multivariate supOU processes., Ann. Appl. Probab. 21 140–182.
  • [11] Barndorff-Nielsen, O. E. and Stelzer, R. (2013). The multivariate supOU stochastic volatility model., Math. Finance 23 275–296.
  • [12] Barndorff-Nielsen, O. E. and Veraart, A. E. D. (2013). Stochastic volatility of volatility and variance risk premia., J. Fin. Econometrics 11 1–46.
  • [13] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987)., Regular Variation. Cambridge University Press, Cambridge.
  • [14] Brandes, D-P. and Curato I. V. (2018). On the sample autocovariance of a Lévy driven moving average process when sampled at a renewal sequence., arxiv:1804.02254v1.
  • [15] Brandes, D-P., Curato I. V. and Stelzer, R. (2019). Preservation of strong mixing and weak dependence under renewal sampling., Work in progress.
  • [16] Brockwell, P.J. (2001). Lévy driven CARMA processes. Non-linear non-Gaussian models and related filtering methods (Tokyo, 2000)., Ann. Inst. Statist. Math. 53 113–124.
  • [17] Cont, R. and Tankov, P. (2004)., Financial Modeling With Jump Processes. Chapman & Hall/CRC, Boca Raton.
  • [18] Dedecker, J. and Doukhan, P. (2003). A new covariance inequality and applications., Stochastic Process Appl. 106 63–80.
  • [19] Dedecker, J., Doukhan, P., Lang, G., Léon, J. R., Louhichi, S. and Prieur, C. (2008)., Weak dependence: with examples and applications. Springer.
  • [20] Dedecker, J. and Rio, E. (2000). On the functional central limit theorem for stationary processes., Ann. Inst. H. Poincaré Probab. Statist. 36 1–34.
  • [21] Doukhan, P. (1994)., Mixing: properties and examples. Lecture Notes in Statistics, 85. Springer-Verlag, New York.
  • [22] Doukhan, P., Fokianos, K. and Li, X. (2012). On weak dependence conditions: the case of discrete valued processes., Statist. Probab. Lett. 82 1941–1948.
  • [23] Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities., Stochastic Process. Appl. 84 313–342.
  • [24] Doukhan, P. and Wintenberger, O. (2007). An invariance principle for weakly dependent stationary general models., Probab. Math. Statist. 27 45–73.
  • [25] Fasen, V. and Klüppelberg, C. (2007). Extremes of supOU processes, in, Stochastic Analysis and Applications: The Abel Symposium 2005, Abel Symposia, vol. 2, edited by F.E. Benth, G. Di Nunno, T. Lindstrom, B. Øksendal and T. Zhang. Springer, Berlin.
  • [26] Fechner, Z. and Stelzer, R. (2018). Limit behavior of the truncated pathwise Fourier-transformation of Lévy driven CARMA processes for non-equidistant discrete time observations., Statist. Sinica 28 1633–1650.
  • [27] Fuchs, F. and Stelzer, R. (2013). Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model., ESAIM Probab. Stat. 17 455–471.
  • [28] Giraitis, L., Koul, H. L. and Surgailis, D. (2012)., Large Sample Inference for Long Memory Processes. Imperial College Press, London.
  • [29] Grahovac, D., Leonenko, N., Sikorskii, A. and Taqqu, M. S. (2018). The unusual properties of aggregated superpositions of Ornstein-Uhlenbeck type processes., Bernoulli, to appear.
  • [30] Grahovac, D., Leonenko, N., Sikorskii, A. and Tešnjak, I. (2016). Intermittency of Superposition of Ornstein-Uhlenbeck type processes., J. Stat. Phys. 165 390–408.
  • [31] Grahovac, D., Leonenko, N. and Taqqu, M. S. (2017). Limit theorems,scaling of moments and intermittency for integrated finite variance supOU processes., ArXiv:1711.09623v1 .
  • [32] Griffin, J. E. and Steel, M. F. (2010). Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein-Uhlenbeck processes., Comput. Statist. Data Anal. 11 2594–2608.
  • [33] Heyde, C. C. and Leonenko, N. N. (2005). Student processes., Adv. in Appl. Probab. 37 342–365.
  • [34] Jacod, J. and Shiryaev, A. N. (2003)., Limit theorems for stochastic processes, 2nd ed. Springer, Berlin.
  • [35] Kelly, B. C., Treu, T., Malkan, M., Pancoast, A. and Woo, J-H. (2013). Active Galactic Nucleus Black Hole Mass Estimates in the Era of Time Domain Astronomy., Astrophys. J. 779 1–18.
  • [36] Kerss, A. D. J., Leonenko, N. N. and Sikorskii, A. (2014). Risky asset models with tempered stable fractal activity time., Stoch. Anal. Appl. 32 642–663.
  • [37] Leonenko, N. N, Petherick, S. and Sikorskii, A. (2012). Fractal activity time models for risky asset with dependence and generalized hyperbolic distributions., Stoch. Anal. Appl. 30 476–492.
  • [38] Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes., Bernoulli 12 1099–1126.
  • [39] Marquardt, T. and Stelzer, R. (2007). Multivariate CARMA process., Stochastic Processes Appl. 117 96–120.
  • [40] Masuda, H. (2007). Ergodicity and exponential $\beta $-mixing bounds for multidimensional diffusions with jumps., Stochastic Process. Appl. 117 35–56.
  • [41] Mátyás, L. (1999)., Generalized method of moments estimation, Vol. 5. Cambridge University Press.
  • [42] Moser, M. and Stelzer, R. (2013). Functional regular variation of Lévy-driven multivariate mixed moving average processes., Extremes 16 351–382.
  • [43] Pedersen, J. (2003). The Lévy-Itô decomposition of an independently scattered random measure., MaPhySto Research Paper 2, available at http://www.maphysto.dk.
  • [44] Pigorsch, C. and Stelzer, R. (2009). A multivariate Ornstein-Uhlenbeck type stochastic volatility model., preprint available on www.uni-ulm.de/en/mawi/finmath/people/stelzer/publications.html.
  • [45] Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes., Probab. Theory and Relat. Fields 82 451–487.
  • [46] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition., Proc. Nat. Acad. Sci. U.S.A. 42 43–47.
  • [47] Sato, K. I. (1999)., Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [48] Stelzer, R., Tossdorf, T. and Wittilinger, M. (2015). Moment based estimation of supOU processes and a related stochastic volatility model., Stat. Risk Model. 32 1–24.
  • [49] Stelzer, R. and Zavĭsin, J. (2015). Derivative pricing under the possibility of long memory in the supOU stochastic volatility model, in, Innovations in Quantitative Risk Management edited by K. Glau, M. Scherer and R. Zagst. Springer.