Theory of Probability and Mathematical Statistics
Weak convergence of weighted additive functionals of long-range dependent fields
T. Alodat, A. Olenko
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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 same distributions under certain assumptions. The result is applied to integrals over a multidimensional rectangle with a constant weight function.
Keywords: Random fields, Hermite-type distributions, non-central limit theorems, longrange dependence, weighted nonlinear functionals.
Bibliography: 1. V. Anh, N. Leonenko, A. Olenko, V. Vaskovych. On rate of convergence in non-central limit theorems, arXiv:1703.05900 (2017).
2. V. Anh, N. Leonenko, A. Olenko. On the rate of convergence to Rosenblatt-type distribution, J. Math. Anal. Appl., 425 (2015), 111-132.
3. S. Bai, M. Taqqu. Multivariate limit theorems in the context of long-range dependence, J. Time Series Anal., 34 (2013), 717-743.
4. J. Beran. Statistics for long-memory processes, CRC press, 1994.
5. N. Bingham, C. Goldie, J. Teugels. Regular variation, Cambridge University Press, 1989.
6. V. Buldygin, Y. Kozachenko, N. Leonenko, Works of M. I. Yadrenko in the theory of random fields, Ukr. Math. J., 441 (1992), 1343-1348.
7. R. Dobrushin, P. Major. Non-central limit theorems for non-linear functional of Gaussian fields, Probab. Theory Related Fields, 50 (1979), 27-52.
8. P. Doukhan, G. Oppenheim, M. Taqqu. Theory and applications of long-range dependence, Springer, 2002.
9. P. Doukhan, M. Taqqu, G. Oppenheim. Long-range dependence: theory and applications,Birkhauser, 2002.
10. A. Ivanov, N. Leonenko. Statistical analysis of random fields, Kluwer Academic Publisher, Dordrecht, 1989.
11. N. Leonenko. Limit theorems for random fields with singular spectrum, Springer, 1999.
12. N. Leonenko, A. Olenko. Sojourn measures of Student and Fisher-Snedecor random fields, Bernoulli, 20 (2014), 1454-1483.
13. N. Leonenko, E. Taufer. Weak convergence of functionals of stationary long memory processes to Rosenblatt-type distributions, J. Statist. Plann. Inference, 136 (2006), 1220-1236.
14. P. Major. Multiple Wiener-Ito Integrals: With Applications to Limit Theorems, Springer-Verlag, 2014.
15. A. Olenko. Limit theorems for weighted functionals of cyclical long-range dependent random fields, Stoch. Anal. Appl., 31 (2013), 199-213.
16. A. Olenko, B. Klykavka. A limit theorem for random fields with a singularity in the spectrum, Theory Probab. Math. Statist., 81 (2010), 147-158.
17. M. Pakkanen, Reveillac. Functional limit theorems for generalized variations of the fractional Brownian sheet, Bernoulli, 22 (2016), 1671-1708.
18. W. Palma. Long-memory time series: theory and methods, John Wiley and Sons, 2007.
19. V. Pipiras, M. Taqqu. Regularization and integral representations of Hermite processes, Statist. Probab. Lett., 80 (2010), 2014-2023.
20. M. Rosenblatt. Limit theorems for Fourier transforms of functionals of Gaussian sequences,Probab. Theory Related Fields, 55 (1981), 123-132.
21. M. Rosenblatt. Independence and dependence, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., 2 (1961), 431-443.
22. M. Taqqu. Convergence of integrated processes of arbitrary Hermite rank, Probab. Theory Related Fields, 50 (1979), 53-83.
23. M. Taqqu. Weak convergence to fractional Brownian motion and to the Rosenblatt process, Probab. Theory Related Fields, 31 (1975), 287-302.
24. M. Yadrenko. Spectral theory of random fields, Optimization Software, New York, 1983.