• Bernoulli
  • Volume 24, Number 4A (2018), 3117-3146.

On limit theory for Lévy semi-stationary processes

Andreas Basse-O’Connor, Claudio Heinrich, and Mark Podolskij

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we present some limit theorems for power variation of Lévy semi-stationary processes in the setting of infill asymptotics. Lévy semi-stationary processes, which are a one-dimensional analogue of ambit fields, are moving average type processes with a multiplicative random component, which is usually referred to as volatility or intermittency. From the mathematical point of view this work extends the asymptotic theory investigated in (Power variation for a class of stationary increments Lévy driven moving averages. Preprint), where the authors derived the limit theory for $k$th order increments of stationary increments Lévy driven moving averages. The asymptotic results turn out to heavily depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal–Getoor index $\beta\in(0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $\alpha$. In this paper, we will study the first order asymptotic theory for Lévy semi-stationary processes with a random volatility/intermittency component and present some statistical applications of the probabilistic results.

Article information

Bernoulli, Volume 24, Number 4A (2018), 3117-3146.

Received: April 2016
Revised: February 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

high frequency data Lévy semi-stationary processes limit theorems power variation stable convergence


Basse-O’Connor, Andreas; Heinrich, Claudio; Podolskij, Mark. On limit theory for Lévy semi-stationary processes. Bernoulli 24 (2018), no. 4A, 3117--3146. doi:10.3150/17-BEJ956.

Export citation


  • [1] Barndorff-Nielsen, O.E. and Basse-O’Connor, A. (2011). Quasi Ornstein–Uhlenbeck processes. Bernoulli 17 916–941.
  • [2] Barndorff-Nielsen, O.E., Benth, F.E. and Veraart, A.E.D. (2013). Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes. Bernoulli 19 803–845.
  • [3] Barndorff-Nielsen, O.E., Corcuera, J.M. and Podolskij, M. (2009). Power variation for Gaussian processes with stationary increments. Stochastic Process. Appl. 119 1845–1865.
  • [4] Barndorff-Nielsen, O.E., Corcuera, J.M. and Podolskij, M. (2011). Multipower variation for Brownian semistationary processes. Bernoulli 17 1159–1194.
  • [5] Barndorff-Nielsen, O.E., Corcuera, J.M. and Podolskij, M. (2013). Limit theorems for functionals of higher order differences of Brownian semi-stationary processes. In Prokhorov and Contemporary Probability Theory (A.N. Shiryaev, S.R.S. Varadhan and E.L. Presman, eds.). Springer Proc. Math. Stat. 33 69–96. Heidelberg: Springer.
  • [6] Barndorff-Nielsen, O.E., Corcuera, J.M., Podolskij, M. and Woerner, J.H.C. (2009). Bipower variation for Gaussian processes with stationary increments. J. Appl. Probab. 46 132–150.
  • [7] Barndorff-Nielsen, O.E., Graversen, S.E., Jacod, J., Podolskij, M. and Shephard, N. (2006). A central limit theorem for realised power and bipower variations of continuous semimartingales. In From Stochastic Calculus to Mathematical Finance (Yu. Kabanov, R. Liptser and J. Stoyanov, eds.) 33–68. Berlin: Springer.
  • [8] Barndorff-Nielsen, O.E., Jensen, E.B.V., Jónsdóttir, K.Y. and Schmiegel, J. (2007). Spatio-temporal modelling – with a view to biological growth. In Statistical Methods for Spatio-Temporal Systems (B. Finkenstädt, L. Held and V. Isham, eds.) 47–75. London: Chapman & Hall/CRC.
  • [9] Barndorff-Nielsen, O.E., Pakkanen, M.S. and Schmiegel, J. (2014). Assessing relative volatility/intermittency/energy dissipation. Electron. J. Stat. 8 1996–2021.
  • [10] Barndorff-Nielsen, O.E. and Schmiegel, J. (2007). Ambit processes: With applications to turbulence and tumour growth. In Stochastic Analysis and Applications. Abel Symp. 2 93–124. Berlin: Springer.
  • [11] Barndorff-Nielsen, O.E. and Schmiegel, J. (2008). Time change, volatility, and turbulence. In Mathematical Control Theory and Finance (A. Sarychev, A. Shiryaev, M. Guerra and M.d.R. Grossinho, eds.) 29–53. Berlin: Springer.
  • [12] Barndorff-Nielsen, O.E. and Schmiegel, J. (2009). Brownian semistationary processes and volatility/intermittency. In Advanced Financial Modelling. Radon Ser. Comput. Appl. Math. 8 1–25. Berlin: Walter de Gruyter.
  • [13] Basse-O’Connor, A., Lachièze-Rey, R. and Podolskij, M. (2016). Power variation for a class of stationary increments Lévy driven moving averages. Ann. Probab. To appear.
  • [14] Basse-O’Connor, A. and Podolskij, M. (2017). On critical cases in limit theory for stationary increments Lévy driven moving averages. Stochastics 89 360–383.
  • [15] Benassi, A., Cohen, S. and Istas, J. (2004). On roughness indices for fractional fields. Bernoulli 10 357–373.
  • [16] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. New York: Wiley.
  • [17] Chronopoulou, A., Viens, F.G. and Tudor, C.A. (2009). Variations and Hurst index estimation for a Rosenblatt process using longer filters. Electron. J. Stat. 3 1393–1435.
  • [18] Coeurjolly, J.-F. (2001). Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 199–227.
  • [19] Dang, T.T.N. and Istas, J. (2015). Estimation of the Hurst and the stability indices of a H-self-similar stable process. Working paper. Available at arXiv:1506.05593.
  • [20] Gärtner, K. and Podolskij, M. (2015). On non-standard limits of Brownian semi-stationary processes. Stochastic Process. Appl. 125 653–677.
  • [21] Grahovac, D., Leonenko, N.N. and Taqqu, M.S. (2015). Scaling properties of the empirical structure function of linear fractional stable motion and estimation of its parameters. J. Stat. Phys. 158 105–119.
  • [22] Guyon, X. and León, J. (1989). Convergence en loi des $H$-variations d’un processus gaussien stationnaire sur $\textbf{R}$. Ann. Inst. Henri Poincaré Probab. Stat. 25 265–282.
  • [23] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
  • [24] Jacod, J. and Protter, P. (2012). Discretization of Processes. Stochastic Modelling and Applied Probability 67. Heidelberg: Springer.
  • [25] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Berlin: Springer.
  • [26] Kwapień, S. and Woyczyński, W.A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Probability and Its Applications. Boston, MA: Birkhäuser.
  • [27] Musielak, J. (1983). Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034. Berlin: Springer.
  • [28] Nourdin, I. and Réveillac, A. (2009). Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$. Ann. Probab. 37 2200–2230.
  • [29] Podolskij, M. and Vetter, M. (2010). Understanding limit theorems for semimartingales: A short survey. Stat. Neerl. 64 329–351.
  • [30] Rajput, B.S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 451–487.
  • [31] Rosiński, J. and Woyczyński, W.A. (1986). On Itô stochastic integration with respect to $p$-stable motion: Inner clock, integrability of sample paths, double and multiple integrals. Ann. Probab. 14 271–286.
  • [32] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
  • [33] Skorohod, A.V. (1956). Limit theorems for stochastic processes. Theory Probab. Appl. 1 261–190.
  • [34] Tudor, C.A. and Viens, F.G. (2009). Variations and estimators for self-similarity parameters via Malliavin calculus. Ann. Probab. 37 2093–2134.
  • [35] Whitt, W. (2002). Stochastic-Process Limits. New York: Springer.