The Annals of Probability

On extremal theory for self-similar processes

J. M. P. Albin

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Abstract

We derive upper and lower asymptotic bounds for the distribution of the supremum for a self-similar stochastic process. As an intermediate step, most proofs relate suprema to sojourns before proceeding to an appropriate discrete approximation.

Our results rely on one or more of three assumptions, which in turn essentially require weak convergence, existence of a first moment and tightness, respectively. When all three assumptions hold, the upper and lower bounds coincide (Corollary 1).

For P-smooth processes, weak convergence can be replaced with the use of a certain upcrossing intensity that works even for (a.s.) discontinuous processes (Theorem 7).

Results on extremes for a self-similar process do not on their own imply results for Lamperti’s associated stationary process or vice versa, but we show that if the associated process satisfies analogues of our three assumptions, then the assumptions hold for the self-similar process itself. Through this connection, new results on extremes for self-similar processes can be derived by invoking the stationarity literature.

Examples of application include Gaussian processes in $\mathbb{R}^n$, totally skewed $\alpha$-stable processes, Kesten–Spitzer processes and Rosenblatt processes.

Article information

Source
Ann. Probab., Volume 26, Number 2 (1998), 743-793.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855649

Digital Object Identifier
doi:10.1214/aop/1022855649

Mathematical Reviews number (MathSciNet)
MR1626515

Zentralblatt MATH identifier
0937.60033

Subjects
Primary: 60G18: Self-similar processes 60G70: Extreme value theory; extremal processes
Secondary: 60F10: Large deviations 60G10: Stationary processes

Keywords
Extremes Lamperti's transformation self-similar process sojourns

Citation

Albin, J. M. P. On extremal theory for self-similar processes. Ann. Probab. 26 (1998), no. 2, 743--793. doi:10.1214/aop/1022855649. https://projecteuclid.org/euclid.aop/1022855649


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References

  • Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Probab. 18 92-128. Albin, J. M. P. (1992a). On the general law of iterated logarithm with application to self-similar processes and to Gaussian processes in n and Hilbert space. Stochastic Process. Appl. 41 1-31. Albin, J. M. P. (1992b). Extremes and crossings for differentiable stationary processes with application to Gaussian processes in m and Hilbert space. Stochastic Process. Appl. 42 119-147.
  • Albin, J. M. P. (1995). Upper and lower classes for L2and Lp-norms of Brownian motion and norms of -stable motion. Stochastic Process. Appl. 58 91-103.
  • Albin, J. M. P. (1997). Extremes for non-anticipating moving averages of totally skewed -stable motion. Statist. Probab. Lett. 36 289-297.
  • Albin, J. M. P. (1998). A note on Rosenblatt distributions. Statist. Probab. Lett. To appear.
  • Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Probab. 10 1-46.
  • Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole, Belmont, CA.
  • Boylan, E. (1964). Local times for a class of Markoff processes. Illinois J. Math. 8 19-39.
  • Cameron, R. H. and Martin, W. T. (1944). The Wiener measure of Hilbert neighborhoods in the space of continuous functions. J. Math. Phys. 23 195-209.
  • Chan, T., Dean, D. S., Jansons, K. M. and Rogers, L. C. G. (1994). On polymer conformations in elongational flows. Comm. Math. Phys. 160 239-257.
  • de Acosta, A. (1977). Asymptotic behavior of stable measures. Ann. Probab. 5 494-499.
  • Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for non-linear functionals of Gaussian fields.Wahrsch. Verw. Gebiete 50 27-52.
  • Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
  • Duplantier, B. (1989). Areas of planar Brownian curves. J. Phys. A Math. Gen. 22 3033-3048.
  • Kasahara, Y., Maejima, M. and Vervaat, W. (1988). Log-fractional stable processes. Stochastic Process. Appl. 30 329-339.
  • Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self similar processes.Wahrsch. Verw. Gebiete 50 5-25.
  • Konstant, D. G. and Pitebarg, V. I. (1993). Extreme values of the cyclostationary Gaussian random process. J. Appl. Probab. 30 82-97.
  • Maejima, M. (1983). On a class of self-similar processes.Wahrsch. Verw. Gebiete 62 235-245.
  • Li, W. V. (1992). Limit theorems for the square integral of Brownian motion and its increments. Stochastic Process. Appl. 41 223-239.
  • Pickands, J., III (1968). Moment convergence of sample extremes. Ann. Math. Statist. 39 881-889.
  • Pickands, J., III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51-73.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Rosenblatt, M. (1961). Independence and dependence. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 431-443. Univ. California Press, Berkeley.
  • Samorodnitsky, G. (1988). Extrema of skewed stable processes. Stochastic Process. Appl. 30 17-39.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance. Chapman and Hall, London.
  • Scheff´e, H. (1947). A useful convergence theorem for probability distributions. Ann. Math. Statist. 18 434-438.
  • Sharpe, K. (1978). Some properties of the crossings process generated by a stationary 2 process. Adv. in Appl. Probab. 10 373-391.
  • Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process.Wahrsch. Verw. Gebiete 31 287-302.
  • Taqqu, M. S. (1986). A bibliographical guide to self-similar processes and long-range dependence. In Dependence in Probability and Statistics. Birkh¨auser, Boston.
  • Taqqu, M. S. and Czado, C. (1985). A survey of functional laws of the iterated logarithm for self-similar processes. Stochastic Models 1 77-115.
  • Taqqu, M. S. and Wolpert, R. (1983). Infinite variance self-similar processes subordinate to a Poisson measure.Wahrsch. Verw. Gebiete 62 53-72.
  • Vervaat, W. (1985). Sample path properties of self-similar processes with stationary increments. Ann. Probab. 13 1-27.
  • Yor, M. (1992). Some Aspects of Brownian Motion. Part I: Some Special Functionals. Birkh¨auser, Boston.