## Electronic Journal of Probability

### On limit theory for functionals of stationary increments Lévy driven moving averages

#### Abstract

In this paper we present new limit theorems for variational functionals of stationary increments Lévy driven moving averages in the high frequency setting. More specifically, we will show the “law of large numbers” and a “central limit theorem”, which heavily rely on the kernel, the driving Lévy process and the properties of the functional under consideration. The first order limit theory consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we prove the associated weak limit theory, which again consists of three different cases. Our work is related to [10, 7], who considered power variation functionals of stationary increments Lévy driven moving averages. However, the asymptotic theory of the present paper is more complex. In particular, the weak limit theorems are derived for an arbitrary Appell rank of the involved functional.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 79, 42 pp.

Dates
Received: 23 July 2018
Accepted: 18 June 2019
First available in Project Euclid: 5 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1567648850

Digital Object Identifier
doi:10.1214/19-EJP336

Mathematical Reviews number (MathSciNet)
MR4003132

Zentralblatt MATH identifier
07107386

#### Citation

Basse-O’Connor, Andreas; Heinrich, Claudio; Podolskij, Mark. On limit theory for functionals of stationary increments Lévy driven moving averages. Electron. J. Probab. 24 (2019), paper no. 79, 42 pp. doi:10.1214/19-EJP336. https://projecteuclid.org/euclid.ejp/1567648850

#### References

• [1] A. Ayache and J. Hamonier, Linear fractional stable motion: A wavelet estimator of the parameter, Stat. Probabil. Lett. 82 (2012), no. 8, 1569–1575.
• [2] J.-M. Bardet and D. Surgailis, Nonparametric estimation of the local Hurst function of multifractional Gaussian processes, Stochastic Process. Appl. 123 (2013), no. 3, 1004–1045.
• [3] O.E. Barndorff-Nielsen, J.M. Corcuera, and M. Podolskij, Power variation for Gaussian processes with stationary increments, Stochastic Process. Appl. 119 (2009), no. 6, 1845–1865.
• [4] O.E. Barndorff-Nielsen, J.M. Corcuera, and M. Podolskij, Multipower variation for Brownian semistationary processes, Bernoulli 17 (2011), no. 4, 1159–1194.
• [5] O.E. Barndorff-Nielsen, J.M. Corcuera, M. Podolskij, and J.H.C. Woerner, Bipower variation for Gaussian processes with stationary increments, J. Appl. Probab. 46 (2009), no. 1, 132–150.
• [6] O.E. Barndorff-Nielsen, S.E. Graversen, J. Jacod, M. Podolskij, and N. Shephard, A central limit theorem for realised power and bipower variations of continuous semimartingales, From stochastic calculus to mathematical finance, Springer, Berlin, 2006, pp. 33–68.
• [7] A. Basse-O’Connor and M. Podolskij, On critical cases in limit theory for stationary increments Lévy driven moving averages, Stochastics 89 (2017), no. 1, 360–383.
• [8] A. Basse-O’Connor and J. Rosiński, On infinitely divisible semimartingales, Probab. Theory Related Fields 164 (2016), no. 1–2, 133–163.
• [9] A. Basse-O’Connor, C. Heinrich, and M. Podolskij, On limit theory for lévy semi-stationary processes, Bernoulli 24 (2018), no. 4A, 3117–3146.
• [10] A. Basse-O’Connor, R. Lachièze-Rey, and M. Podolskij, Power variation for a class of stationary increments Lévy driven moving averages, Ann. Prob. 45 (2017), no. 6B, 4477–4528.
• [11] A. Benassi, S. Cohen, and J. Istas, On roughness indices for fractional fields, Bernoulli 10 (2004), no. 2, 357–373.
• [12] K.N. Berk, A central limit theorem for $m$-dependent random variables with unbounded $m$, Ann. Probab. 1 (1973), 352–354.
• [13] P. Billingsley, Convergence of probability measures, second ed., John Wiley & Sons, Inc., New York, 1999.
• [14] M. Braverman and G. Samorodnitsky, Symmetric infinitely divisible processes with sample paths in Orlicz spaces and absolute continuity of infinitely divisible processes, Stochastic Process. Appl. 78 (1998), no. 1, 1–26.
• [15] S. Cambanis, C.D. Hardin, Jr., and A. Weron, Ergodic properties of stationary stable processes, Stochastic Process. Appl. 24 (1987), no. 1, 1–18.
• [16] S. Cohen, C. Lacaux, and M. Ledoux, A general framework for simulation of fractional fields, Stochastic Process. Appl. 118 (2008), no. 9, 1489–1517.
• [17] T.T.N. Dang and J. Istas, Estimation of the Hurst and the stability indices of a $H$-self-similar stable process, Electron. J. Stat. 11 (2017), no. 2, 4103–4150.
• [18] S. Glaser, A law of large numbers for the power variation of fractional Lévy processes, Stoch. Anal. Appl. 33 (2015), no. 1, 1–20.
• [19] X. Guyon and J. León, Convergence en loi des $H$-variations d’un processus gaussien stationnaire sur $\mathbf{R}$, Ann. I. H. Poincaré (B). 25 (1989), no. 3, 265–282.
• [20] E. Häusler and H. Luschgy, Stable convergence and stable limit theorems, Springer, Cham, 2015.
• [21] H. Ho and T. Hsing, Limit theorems for functionals of moving averages, Ann. Probab. 25 (1997), no. 4, 1636–1669.
• [22] J. Jacod, Asymptotic properties of realized power variations and related functionals of semimartingales, Stochastic Process. Appl. 118 (2008), no. 4, 517–559.
• [23] J. Jacod and P. Protter, Discretization of processes, Springer, Heidelberg, 2012.
• [24] O. Kallenberg, Foundations of modern probability, second ed., Springer, Heidelberg, 2002.
• [25] J. Lebovits and M. Podolskij, Estimation of the global regularity of a multifractional Brownian motion, Electron. J. Stat. 11 (2017), no. 1, 78–98.
• [26] S. Mazur, D. Otryakhin, and M. Podolskij, Estimation of the linear fractional stable motion, Bernoulli (2018, accepted), Preprint at arXiv:1802.06373 [stat.ME].
• [27] V. Pipiras and M.S. Taqqu, Central limit theorems for partial sums of bounded functionals of infinite-variance moving averages, Bernoulli 9 (2003), no. 5, 833–855.
• [28] V. Pipiras and M.S. Taqqu, Long-range dependence and self-similarity, vol. 45, Cambridge University Press, Cambridge, 2017.
• [29] V. Pipiras, M.S. Taqqu, and P. Abry, Bounds for the covariance of functions of infinite variance stable random variables with applications to central limit theorems and wavelet-based estimation, Bernoulli 13 (2007), no. 4, 1091–1123.
• [30] B.S. Rajput and J. Rosiński, Spectral representations of infinitely divisible processes, Probab. Theory Rel. 82 (1989), no. 3, 451–487.
• [31] A. Rényi, On stable sequences of events, Sankhyā Ser. A 25 (1963), 293–302.
• [32] G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian random processes, Chapman & Hall, New York, 1994.
• [33] K. Sato, Lévy processes and infinitely divisible distributions, Cambridge University Press, Cambridge, 2013.
• [34] A.V. Skorohod, Limit theorems for stochastic processes, Teor. Ver. Prim. 1 (1956), 289–319.
• [35] D. Surgailis, Stable limits of empirical processes of moving averages with infinite variance, Stochastic Process. Appl. 100 (2002), no. 1–2, 255–274.
• [36] D. Surgailis, Stable limits of sums of bounded functions of long-memory moving averages with finite variance, Bernoulli 10 (2004), no. 2, 327–355.
• [37] B. von Bahr and C.G. Esseen, Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq 2$, Ann. Math. Stat. 36 (1965), 299–303.
• [38] T. Watanabe, Asymptotic estimates of multi-dimensional stable densities and their applications, Trans. Amer. Math. Soc. 359 (2007), no. 6, 2851–2879.
• [39] W. Whitt, Stochastic-process limits, Springer-Verlag, New York, 2002.
• [40] V. M. Zolotarev, One-dimensional stable distributions, vol. 65, American Mathematical Society, Providence, RI, 1986.