## The Annals of Probability

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

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 18 May 2004

https://projecteuclid.org/euclid.aop/1084884857

Digital Object Identifier
doi:10.1214/009117904000000261

Mathematical Reviews number (MathSciNet)
MR2060304

Zentralblatt MATH identifier
1049.60027

#### Citation

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. https://projecteuclid.org/euclid.aop/1084884857

#### References

• 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.