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

  • Ayan Bhattacharya (a1) and Parthanil Roy (a2)

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.

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Corresponding author

* 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.

References

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[1]Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory (Math. Surveys Monogr. 50). American Mathematical Society, Providence, RI.
[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, 659672.
[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, 921934.
[4]Cappé, O.et al. (2002). Long-range dependence and heavy-tail modeling for teletraffic data. IEEE Signal Process. Magazine 19, 1427.
[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 559579.
[6]Fasen, V. and Roy, P. (2016). Stable random fields, point processes and large deviations. Stoch. Process. Appl. 126, 832856.
[7]Giraitis, L. and Taqqu, M. S. (1999). Whittle estimator for finite-variance non-Gaussian time series with long memory. Ann. Statist. 27, 178203.
[8]Hurst, H. E. (1951). Long-term storage capacity of reservoirs. Trans. Amer. Soc. Civil Eng. 116, 770799.
[9]Hurst, H. E. (1956). Methods of using long-term storage in reservoirs. Proc. Inst. Civil Eng. 5, 519543.
[10]Karcher, W. and Spodarev, E. (2011). Kernel function estimation for stable moving average random fields. Unpublished manuscript.
[11]Karcher, W., Shmileva, E. and Spodarev, E. (2013). Extrapolation of stable random fields. J. Multivariate Anal. 115, 516536.
[12]Lang, S. (2002). Algebra Revised, 3rd edn. Springer, New York.
[13]Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
[14]Mikosch, T. and Samorodnitsky, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Prob. 28, 18141851.
[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, 217228.
[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.
[17]Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
[18]Resnick, S. and Samorodnitsky, G. (2004). Point processes associated with stationary stable processes. Stoch. Process. Appl. 114, 191209.
[19]Robinson, P. M. (1995). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23, 10481072.
[20]Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Prob. 23, 11631187.
[21]Rosiński, J. (2000). Decomposition of stationary α-stable random fields. Ann. Prob. 28, 17971813.
[22]Roy, P. (2010). Ergodic theory, abelian groups and point processes induced by stable random fields. Ann. Prob. 38, 770793.
[23]Roy, P. and Samorodnitsky, G. (2008). Stationary symmetric α-stable discrete parameter random fields. J. Theoret. Prob. 21, 212233.
[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, 14381468.
[25]Samorodnitsky, G. (2006). Long range dependence. Found. Trends Stoch. Syst. 1, 163257.
[26]Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.
[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. 6187.
[28]Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1993). Stable mixed moving averages. Prob. Theory Relat. Fields 97, 543558.
[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. 379392.

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

  • Ayan Bhattacharya (a1) and Parthanil Roy (a2)

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