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ISSN 1547-7363(online) ISSN 0094-9000(print)
Weak convergence of weighted additive functionals of long-range dependent fields
Authors:
T. Alodat and A. Olenko
Translated by:
The authors
Journal:
Theor. Probability and Math. Statist. 97 (2018), 1-16
MSC (2010):
Primary 60G60; Secondary 60G12, 60F05
DOI:
https://doi.org/10.1090/tpms/1044
Published electronically:
February 21, 2019
MathSciNet review:
3745995
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We provide asymptotic results for the distribution of weighted nonlinear functionals of Gaussian field with long-range dependence. We also show that integral functionals and the corresponding additive functionals have the same distributions under certain assumptions. The result is applied to integrals over a multidimensional rectangle with a constant weight function.
- V. Anh, N. Leonenko, A. Olenko, and V. Vaskovych, On rate of convergence in non-central limit theorems, Bernoulli (to appear).
- Vo Anh, Nikolai Leonenko, and Andriy Olenko, On the rate of convergence to Rosenblatt-type distribution, J. Math. Anal. Appl. 425 (2015), no. 1, 111–132. MR 3299653, DOI https://doi.org/10.1016/j.jmaa.2014.12.016
- Shuyang Bai and Murad S. Taqqu, Multivariate limit theorems in the context of long-range dependence, J. Time Series Anal. 34 (2013), no. 6, 717–743. MR 3127215, DOI https://doi.org/10.1111/jtsa.12046
- Jan Beran, Statistics for long-memory processes, Monographs on Statistics and Applied Probability, vol. 61, Chapman and Hall, New York, 1994. MR 1304490
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1989. MR 1015093
- V. V. Buldygin, Yu. V. Kozachenko, and N. N. Leonenko, On the works of M. I. Yadrenko in the theory of random fields, Ukraïn. Mat. Zh. 44 (1992), no. 11, 1460–1465 (Russian, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 44 (1992), no. 11, 1343–1348 (1993). MR 1213887, DOI https://doi.org/10.1007/BF01071506
- R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussian fields, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 27–52. MR 550122, DOI https://doi.org/10.1007/BF00535673
- P. Doukhan, G. Oppenheim, and M. Taqqu, Theory and applications of long-range dependence, Springer, 2002.
- Murad S. Taqqu, Fractional Brownian motion and long-range dependence, Theory and applications of long-range dependence, Birkhäuser Boston, Boston, MA, 2003, pp. 5–38. MR 1956042
- A. V. Ivanov and N. N. Leonenko, Statistical analysis of random fields, Mathematics and its Applications (Soviet Series), vol. 28, Kluwer Academic Publishers Group, Dordrecht, 1989. With a preface by A. V. Skorokhod; Translated from the Russian by A. I. Kochubinskiĭ. MR 1009786
- Nikolai Leonenko, Limit theorems for random fields with singular spectrum, Mathematics and its Applications, vol. 465, Kluwer Academic Publishers, Dordrecht, 1999. MR 1687092
- Nikolai Leonenko and Andriy Olenko, Sojourn measures of Student and Fisher-Snedecor random fields, Bernoulli 20 (2014), no. 3, 1454–1483. MR 3217450, DOI https://doi.org/10.3150/13-BEJ529
- Nikolai N. Leonenko and Emanuele Taufer, Weak convergence of functionals of stationary long memory processes to Rosenblatt-type distributions, J. Statist. Plann. Inference 136 (2006), no. 4, 1220–1236. MR 2253760, DOI https://doi.org/10.1016/j.jspi.2004.09.011
- Péter Major, Multiple Wiener-Itô integrals, 2nd ed., Lecture Notes in Mathematics, vol. 849, Springer, Cham, 2014. With applications to limit theorems. MR 3155040
- Andriy Olenko, Limit theorems for weighted functionals of cyclical long-range dependent random fields, Stoch. Anal. Appl. 31 (2013), no. 2, 199–213. MR 3021486, DOI https://doi.org/10.1080/07362994.2013.741410
- A. Ya. Olenko and B. M. Klikavka, A limit theorem for fields with a singularity in the spectrum, Teor. Ĭmovīr. Mat. Stat. 81 (2009), 128–138 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 81 (2010), 147–158. MR 2667316, DOI https://doi.org/10.1090/S0094-9000-2011-00816-2
- Mikko S. Pakkanen and Anthony Réveillac, Functional limit theorems for generalized variations of the fractional Brownian sheet, Bernoulli 22 (2016), no. 3, 1671–1708. MR 3474829, DOI https://doi.org/10.3150/15-BEJ707
- Wilfredo Palma, Long-memory time series, Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2007. Theory and methods. MR 2297359
- Vladas Pipiras and Murad S. Taqqu, Regularization and integral representations of Hermite processes, Statist. Probab. Lett. 80 (2010), no. 23-24, 2014–2023. MR 2734275, DOI https://doi.org/10.1016/j.spl.2010.09.008
- M. Rosenblatt, Limit theorems for Fourier transforms of functionals of Gaussian sequences, Z. Wahrsch. Verw. Gebiete 55 (1981), no. 2, 123–132. MR 608012, DOI https://doi.org/10.1007/BF00535155
- M. Rosenblatt, Independence and dependence, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, pp. 431–443. MR 0133863
- Murad S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 53–83. MR 550123, DOI https://doi.org/10.1007/BF00535674
- Murad S. Taqqu, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 287–302. MR 400329, DOI https://doi.org/10.1007/BF00532868
- M. Ĭ. Yadrenko, Spectral theory of random fields, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York, 1983. Translated from the Russian. MR 697386
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Additional Information
T. Alodat
Affiliation:
Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia
Email:
Alodat.t@students.latrobe.edu.au
A. Olenko
Affiliation:
Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia
Email:
a.olenko@latrobe.edu.au
Keywords:
Random fields,
Hermite-type distributions,
noncentral limit theorems,
long-range dependence,
weighted nonlinear functionals
Received by editor(s):
September 19, 2017
Published electronically:
February 21, 2019
Additional Notes:
Supported in part under the Australian Research Council’s Discovery Project funding scheme (project number DP160101366) and in part by the La Trobe University DRP Grant in Mathematical and Computing Sciences.
Dedicated:
This paper is dedicated to the 85th anniversary of Professor M. Yadrenko’s birth
Article copyright:
© Copyright 2019
American Mathematical Society