Asymptotic theory for quadratic variation of harmonizable fractional stable processes
Authors:
Andreas Basse-O’Connor and Mark Podolskij
Journal:
Theor. Probability and Math. Statist. 110 (2024), 3-12
MSC (2020):
Primary 60F05, 60F15, 60G22, 60G48, 60H05
DOI:
https://doi.org/10.1090/tpms/1203
Published electronically:
May 10, 2024
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Additional Information
Abstract: In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional $\alpha$-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a Lévy-driven Rosenblatt random variable when the Hurst parameter satisfies $H\in (1/2,1)$ and $\alpha (1-H)<1/2$. This result complements the asymptotic theory for fractional stable processes.
References
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References
- A. Ayache and J. Hamonier, Linear fractional stable motion: A wavelet estimator of the $\alpha$ parameter, Statist. Probab. Lett. 82 (2012), 1569–1575. MR 2930661
- A. Ayache and Y. Xiao, Harmonizable fractional stable fields: Local nondeterminism and joint continuity of the local times, Stochastic Process. Appl. 126 (2016), no. 1, 171–185. MR 3426515
- A. Basse-O’Connor, T. Grønbæk, and M. Podolskij, Local asymptotic self-similarity for heavy-tailed harmonizable fractional Lévy motions, ESAIM Probab. Stat. 25 (2021), 286–297. MR 4281739
- A. Basse-O’Connor, C. Heinrich, and M. Podolskij, On limit theory for functionals of stationary increments Lévy driven moving averages, Electron. J. Probab. 24 (2019), no. 79, 1–42. MR 4003132
- A. Basse-O’Connor, R. Lachièze-Rey, and M. Podolskij, Power variation for a class of stationary increments Lévy driven moving averages, Ann. Probab. 45 (2017), no. 6B, 4477–4528. MR 3737916
- A. Basse-O’Connor and M. Podolskij, On critical cases in limit theory for stationary increments Lévy driven moving averages, Festschrift for Bernt Øksendal, Stochastics 81 (2017), no. 1, 360–383. MR 3574707
- A. Basse-O’Connor, M. Podolskij, and C. Thäle, A Berry–Esseén theorem for partial sums of functionals of heavy-tailed moving averages, Electron. J. Probab. 25 (2020), no. 31, 1–31. MR 4073692
- A. Benassi, S. Cohen. and J. Istas, Identification and properties of real harmonizable fractional Lévy motions, Bernoulli 8 (2002), no. 1, 97–115. MR 1884160
- S. Cambanis, C. D. Hardin, and A. Weron, Ergodic properties of stationary stable processes, Stochastic Process. Appl. 24 (1987), 1–18. MR 883599
- S. Cambanis and M. Maejima, Two classes of self-similar stable processes with stationary increments, Stochastic Process. Appl. 32 (1989), no. 2, 305–329. MR 1014456
- T. T. N. Dang and J. Istas, Estimation of the Hurst and the stability indices of a $H$-self-similar stable process, Electron. J. Probab. 11 (2017), 4103–4150. MR 3715823
- M. Dozzi and G. Shevchenko, Real harmonizable multifractional stable process and its local properties, Stochastic Process. Appl. 121 (2011), no. 7, 1509–1523. MR 2802463
- D. Grahovac, N. N. Leonenko and M. S. Taqqu, Scaling properties of the empirical structure function of linear fractional stable motion and estimation of its parameters, J. Stat. Phys. 158 (2015), 105–119. MR 3296276
- J. Hoffmann-Jørgensen, Probability with a view toward statistics, vol. I, Chapman & Hall, New York, 1994. MR 1278485
- O. Kallenberg and J. Szulga, Multiple integration with respect to Poisson and Lévy processes, Probab. Theory and Related Fields. 83 (1989), no. 1-2, 101–134. MR 1012497
- N. Kôno and M. Maejima, Hölder continuity of sample paths of some self-similar stable processes, Tokyo J. Math. 14 (1991), no. 1, 93–100. MR 1108158
- S. Kwapien and W. A. Woyczynski, Double stochastic integrals, random quadratic forms and random series in Orlicz spaces, Ann. Probab. 15 (1987), no. 3, 1072–1096. MR 893915
- M. M. Ljungdahl and M. Podolskij, A limit theorem for a class of stationary increments Lévy moving average process with multiple singularities, Mod. Stoch. Theory Appl. 5 (2018), no. 3, 297–316. MR 3868543
- M. M. Ljungdahl and M. Podolskij, A minimal contrast estimator for the linear fractional stable motion, Stat. Inference Stoch. Process. 23 (2020), 381–413. MR 4123929
- M. M. Ljungdahl and M. Podolskij, Multidimensional parameter estimation of heavy-tailed moving averages, Scand. J. Stat. 49 (2022), no. 2, 593–624. MR 4428498
- B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422–437. MR 242239
- S. Mazur, D. Otryakhin, and M. Podolskij, Estimation of the linear fractional stable motion, Bernoulli 26 (2020), no. 1, 226–252. MR 4036033
- V. Pipiras and M. S. Taqqu, Central limit theorems for partial sums of bounded functionals of infinite-variance moving averages, Bernoulli 9 (2003), 833–855. MR 2047688
- 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. MR 2364228
- J. Rosinski, On the structure of stationary stable processes, Ann. Probab. 23 (1995), no. 3, 1163–1187. MR 1349166
- G. Samorodnitsky and M. Taqqu, Multiple stable integrals of Banach-valued functions, J. Theoret. Probab. 3 (1990), 267–287. MR 1046334
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Additional Information
Andreas Basse-O’Connor
Affiliation:
Department of Mathematics, University of Aarhus, DK-8000 Aarhus, Denmark
Email:
basse@math.au.dk
Mark Podolskij
Affiliation:
Department of Mathematics, University of Luxembourg, Luxembourg
Email:
mark.podolskij@uni.lu
Keywords:
Fractional processes,
harmonizable processes,
limit theorems,
quadratic variation,
stable Lévy motion
Received by editor(s):
February 28, 2023
Accepted for publication:
June 29, 2023
Published electronically:
May 10, 2024
Additional Notes:
The second author gratefully acknowledges financial support of ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High Dimensional Diffusions”.
Article copyright:
© Copyright 2024
Taras Shevchenko National University of Kyiv
