Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T18:39:27.622Z Has data issue: false hasContentIssue false
  • English
  • Français

A large sample test for the length of memory of stationary symmetric stable random fields via nonsingular ℤd-actions

Part of: Multivariate analysis Stochastic processes Ergodic theory

Published online by Cambridge University Press:  28 March 2018

Ayan Bhattacharya  and
Parthanil Roy
Ayan Bhattacharya*
Affiliation:
CWI, Amsterdam
Parthanil Roy*
Affiliation:
ISI, Bangalore
*
* Postal address: Stochastics group, CWI, Amsterdam, North Holland, 1098XG, Netherlands. Email address: ayanbhattacharya.isi@gmail.com
** Postal address: Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bangalore, 560059, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

Based on the ratio of two block maxima, we propose a large sample test for the length of memory of a stationary symmetric α-stable discrete parameter random field. We show that the power function converges to 1 as the sample-size increases to ∞ under various classes of alternatives having longer memory in the sense of Samorodnitsky (2004). Ergodic theory of nonsingular ℤd-actions plays a very important role in the design and analysis of our large sample test.

Keywords

Long range dependencestationary SαS random fieldnonsingular group actionextreme value theorystatistical hypothesis testing

MSC classification

Primary: 60G10: Stationary processes 60G60: Random fields 62H15: Hypothesis testing
Secondary: 37A40: Nonsingular (and infinite-measure preserving) transformations
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 179 - 195
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory (Math. Surveys Monogr. 50). American Mathematical Society, Providence, RI. Google Scholar
[2]Beran, J. (1995). Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models. J. R. Statist. Soc. B 57, 659–672. Google Scholar
[3]Beran, J., Bhansali, R. J. and Ocker, D. (1998). On unified model selection for stationary and nonstationary short and long-memory autoregressive processes. Biometrika 85, 921–934. CrossRefGoogle Scholar
[4]Cappé, O.et al. (2002). Long-range dependence and heavy-tail modeling for teletraffic data. IEEE Signal Process. Magazine 19, 14–27. CrossRefGoogle Scholar
[5]Conti, P. L., De Giovanni, L., Stoev, S. A. and Taqqu, M. S. (2008). Confidence intervals for the long memory parameter based on wavelets and resampling. Statistica Sinica 559–579. Google Scholar
[6]Fasen, V. and Roy, P. (2016). Stable random fields, point processes and large deviations. Stoch. Process. Appl. 126, 832–856. Google Scholar
[7]Giraitis, L. and Taqqu, M. S. (1999). Whittle estimator for finite-variance non-Gaussian time series with long memory. Ann. Statist. 27, 178–203. Google Scholar
[8]Hurst, H. E. (1951). Long-term storage capacity of reservoirs. Trans. Amer. Soc. Civil Eng. 116, 770–799. CrossRefGoogle Scholar
[9]Hurst, H. E. (1956). Methods of using long-term storage in reservoirs. Proc. Inst. Civil Eng. 5, 519–543. Google Scholar
[10]Karcher, W. and Spodarev, E. (2011). Kernel function estimation for stable moving average random fields. Unpublished manuscript. Google Scholar
[11]Karcher, W., Shmileva, E. and Spodarev, E. (2013). Extrapolation of stable random fields. J. Multivariate Anal. 115, 516–536. Google Scholar
[12]Lang, S. (2002). Algebra Revised, 3rd edn. Springer, New York. Google Scholar
[13]Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York. Google Scholar
[14]Mikosch, T. and Samorodnitsky, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Prob. 28, 1814–1851. Google Scholar
[15]Montanari, A., Taqqu, M. S. and Teverovsky, V. (1999). Estimating long-range dependence in the presence of periodicity: an empirical study. Math. Comput. Modelling 29, 217–228. CrossRefGoogle Scholar
[16]Panigrahi, S., Roy, P. and Xiao, Y. (2017). Maximal moments and uniform modulus of continuity for stable random fields. Preprint. Available at https://arxiv.org/abs/1709.07135. Google Scholar
[17]Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. CrossRefGoogle Scholar
[18]Resnick, S. and Samorodnitsky, G. (2004). Point processes associated with stationary stable processes. Stoch. Process. Appl. 114, 191–209. CrossRefGoogle Scholar
[19]Robinson, P. M. (1995). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23, 1048–1072. CrossRefGoogle Scholar
[20]Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Prob. 23, 1163–1187. Google Scholar
[21]Rosiński, J. (2000). Decomposition of stationary α-stable random fields. Ann. Prob. 28, 1797–1813. Google Scholar
[22]Roy, P. (2010). Ergodic theory, abelian groups and point processes induced by stable random fields. Ann. Prob. 38, 770–793. CrossRefGoogle Scholar
[23]Roy, P. and Samorodnitsky, G. (2008). Stationary symmetric α-stable discrete parameter random fields. J. Theoret. Prob. 21, 212–233. Google Scholar
[24]Samorodnitsky, G. (2004). Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes. Ann. Prob. 32, 1438–1468. Google Scholar
[25]Samorodnitsky, G. (2006). Long range dependence. Found. Trends Stoch. Syst. 1, 163–257. CrossRefGoogle Scholar
[26]Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York. Google Scholar
[27]Stoev, S. and Taqqu, M. S. (2003). Wavelet estimation for the Hurst parameter in stable processes. In Processes with Long-Range Correlations, Springer, Berlin, pp. 61–87. Google Scholar
[28]Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1993). Stable mixed moving averages. Prob. Theory Relat. Fields 97, 543–558. Google Scholar
[29]Weron, A. and Weron, R. (1995). Computer simulation of Lévy α-stable variables and processes. In Chaos—The Interplay Between Stochastic and Deterministic Behaviour, Springer, Berlin, pp. 379–392. Google Scholar