The Annals of Probability

Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes

Gennady Samorodnitsky

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We study the partial maxima of stationary α-stable processes. We relate their asymptotic behavior to the ergodic theoretical properties of the flow. We observe a sharp change in the asymptotic behavior of the sequence of partial maxima as flow changes from being dissipative to being conservative, and argue that this may indicate a change from a short memory process to a long memory process.

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Ann. Probab., Volume 32, Number 2 (2004), 1438-1468.

First available in Project Euclid: 18 May 2004

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Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 37A40: Nonsingular (and infinite-measure preserving) transformations

Stable process stationary process long memory long range dependence ergodic theory maxima extreme value theory nonsingular flow dissipative flow conservative flow


Samorodnitsky, Gennady. Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes. Ann. Probab. 32 (2004), no. 2, 1438--1468. doi:10.1214/009117904000000261.

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  • Araujo, A. and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.
  • Astrauskas, A., Levy, J. and Taqqu, M. S. (1991). The asymptotic dependence structure of the linear fractional Lévy motion. Liet. Mat. Rink. (Lithuanian Math. J.) 31 1--28.
  • Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.
  • Billingsley, P. (1965). Ergodic Theory and Information. Wiley, New York.
  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  • Cox, D. (1984). Long-range dependence: A review. In Statistics: An Appraisal (H. David and H. David, eds.) 55--74. Iowa State Univ. Press, Ames.
  • Harris, T. and Robbins, H. (1953). Ergodic theory of Markov chains admitting an infinite invariant measure. Proc. Natl. Acad. Sci. 39 860--864.
  • Krengel, U. (1985). Ergodic Theorems. de Gruyter, Berlin.
  • Leadbetter, M., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Maharam, D. (1964). Incompressible transformations. Fund. Math. 56 35--50.
  • Mandelbrot, B. (1975). Limit theorems on the self-normalized range for weakly and strongly dependent processes. Z. Wahrsch. Verw. Gebiete 31 271--285.
  • Mandelbrot, B. and Taqqu, M. (1979). Robust R$/$S analysis of long-run serial correlation. In Proceedings of the 42nd Session of the International Statistical Institute 48 69--104. Amsterdam, Netherlands.
  • Marcus, M. (1984). Extreme values for sequences of stable random variables. In Statistical Extremes and Applications. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 131 311--324. Reidel, Dordrecht.
  • Mikosch, T. and Samorodnitsky, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Probab. 28 1814--1851.
  • Resnick, S. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston.
  • Resnick, S., Samorodnitsky, G. and Xue, F. (2000). Growth rates of sample covariances of stationary symmetric $\alpha$-stable processes associated with null recurrent Markov chains. Stochastic Process. Appl. 85 321--339.
  • Rosiński, J. (1994). Uniqueness of spectral representations of skewed stable processes and stationarity. In Stochastic Analysis on Infinite-Dimensional Spaces (H. Kunita and H.-H. Kuo, eds.) 264--273. Longman, Harlow.
  • Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23 1163--1187.
  • Rosiński, J. and Samorodnitsky, G. (1996). Classes of mixing stable processes. Bernoulli 2 365--378.
  • Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
  • Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1993). Stable mixed moving averages. Probab. Theory Related Fields 97 543--558.
  • Zolotarev, V. (1957). Mellin--Stiltjes transform in probability theory. Theory Probab. Appl. 2 433--460.