## Bernoulli

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

### On limit theory for Lévy semi-stationary processes

#### Abstract

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

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

Dates
Revised: February 2017
First available in Project Euclid: 26 March 2018

https://projecteuclid.org/euclid.bj/1522051235

Digital Object Identifier
doi:10.3150/17-BEJ956

Mathematical Reviews number (MathSciNet)
MR3779712

Zentralblatt MATH identifier
06853275

#### Citation

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. https://projecteuclid.org/euclid.bj/1522051235

#### References

• [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.