Theory of Probability and Mathematical Statistics
Asymptotic theory for quadratic variation of harmonizable fractional stable processes
Andreas Basse-O’Connor and Mark Podolskij
Link
Abstract: In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional -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∈(1/2,1) and α(1-H)<1/2. This result complements the asymptotic theory for fractional stable processes.
Keywords: Fractional processes, harmonizable processes, limit theorems, quadratic variation, stable Lévy motion
Bibliography: Antoine Ayache and Julien Hamonier, Linear fractional stable motion: a wavelet estimator of the αparameter, Statist. Probab. Lett. 82 (2012), no. 8, 1569–1575. MR 2930661, DOI 10.1016/j.spl.2012.04.005
Antoine Ayache and Yimin 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, DOI 10.1016/j.spa.2015.08.001
Andreas Basse-O’Connor, Thorbjørn Grønbæk, and Mark Podolskij, Local asymptotic self-similarity for heavy tailed harmonizable fractional Lévy motions, ESAIM Probab. Stat. 25 (2021), 286–297. MR 4281739, DOI 10.1051/ps/2021011
Andreas Basse-O’Connor, Claudio Heinrich, and Mark Podolskij, On limit theory for functionals of stationary increments Lévy driven moving averages, Electron. J. Probab. 24 (2019), Paper No. 79, 42. MR 4003132, DOI 10.1214/19-ejp336
Andreas Basse-O’Connor, Raphaël Lachièze-Rey, and Mark Podolskij, Power variation for a class of stationary increments Lévy driven moving averages, Ann. Probab. 45 (2017), no. 6B, 4477–4528. MR 3737916, DOI 10.1214/16-AOP1170
Andreas Basse-O’Connor and Mark Podolskij, On critical cases in limit theory for stationary increments Lévy driven moving averages, Stochastics 89 (2017), no. 1, 360–383. MR 3574707, DOI 10.1080/17442508.2016.1191493
Andreas Basse-O’Connor, Mark Podolskij, and Christoph Thäle, A Berry-Esseén theorem for partial sums of functionals of heavy-tailed moving averages, Electron. J. Probab. 25 (2020), Paper No. 31, 31. MR 4073692, DOI 10.1214/20-ejp435
Albert Benassi, Serge Cohen, and Jacques Istas, Identification and properties of real harmonizable fractional Lévy motions, Bernoulli 8 (2002), no. 1, 97–115. MR 1884160
Stamatis Cambanis, Clyde D. Hardin Jr., and Aleksander Weron, Ergodic properties of stationary stable processes, Stochastic Process. Appl. 24 (1987), no. 1, 1–18. MR 883599, DOI 10.1016/0304-4149(87)90024-X
Stamatis Cambanis and Makoto Maejima, Two classes of self-similar stable processes with stationary increments, Stochastic Process. Appl. 32 (1989), no. 2, 305–329. MR 1014456, DOI 10.1016/0304-4149(89)90082-3
Thi To Nhu Dang and Jacques 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. MR 3715823, DOI 10.1214/17-EJS1357
Marco Dozzi and Georgiy Shevchenko, Real harmonizable multifractional stable process and its local properties, Stochastic Process. Appl. 121 (2011), no. 7, 1509–1523. MR 2802463, DOI 10.1016/j.spa.2011.03.012
Danijel Grahovac, Nikolai N. Leonenko, and Murad S. Taqqu, Scaling properties of the empirical structure function of linear fractional stable motion and estimation of its parameters, J. Stat. Phys. 158 (2015), no. 1, 105–119. MR 3296276, DOI 10.1007/s10955-014-1126-4
J. Hoffmann-Jørgensen, Probability with a view toward statistics. Vol. I, Chapman & Hall Probability Series, Chapman & Hall, New York, 1994. MR 1278485, DOI 10.1007/978-1-4899-3019-4
O. Kallenberg and J. Szulga, Multiple integration with respect to Poisson and Lévy processes, Probab. Theory Related Fields 83 (1989), no. 1-2, 101–134. MR 1012497, DOI 10.1007/BF00333146
Norio Kôno and Makoto Maejima, Hölder continuity of sample paths of some self-similar stable processes, Tokyo J. Math. 14 (1991), no. 1, 93–100. MR 1108158, DOI 10.3836/tjm/1270130491
Stanisław Kwapień and Wojbor A. Woyczyński, Double stochastic integrals, random quadratic forms and random series in Orlicz spaces, Ann. Probab. 15 (1987), no. 3, 1072–1096. MR 893915
Mathias Mørck Ljungdahl and Mark 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, DOI 10.15559/18-vmsta111
Mathias Mørck Ljungdahl and Mark Podolskij, A minimal contrast estimator for the linear fractional stable motion, Stat. Inference Stoch. Process. 23 (2020), no. 2, 381–413. MR 4123929, DOI 10.1007/s11203-020-09216-2
Mathias Mørck Ljungdahl and Mark Podolskij, Multidimensional parameter estimation of heavy-tailed moving averages, Scand. J. Stat. 49 (2022), no. 2, 593–624. MR 4428498, DOI 10.1111/sjos.12527
Benoit B. Mandelbrot and John W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422–437. MR 242239, DOI 10.1137/1010093
Stepan Mazur, Dmitry Otryakhin, and Mark Podolskij, Estimation of the linear fractional stable motion, Bernoulli 26 (2020), no. 1, 226–252. MR 4036033, DOI 10.3150/19-BEJ1124
Vladas Pipiras and Murad S. Taqqu, Central limit theorems for partial sums of bounded functionals of infinite-variance moving averages, Bernoulli 9 (2003), no. 5, 833–855. MR 2047688, DOI 10.3150/bj/1066418880
Vladas Pipiras, Murad S. Taqqu, and Patrice 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, DOI 10.3150/07-BEJ6143
Jan Rosiński, On the structure of stationary stable processes, Ann. Probab. 23 (1995), no. 3, 1163–1187. MR 1349166
Gennady Samorodnitsky and Murad S. Taqqu, Multiple stable integrals of Banach-valued functions, J. Theoret. Probab. 3 (1990), no. 2, 267–287. MR 1046334, DOI 10.1007/BF01045162
