The Annals of Probability

On extremal theory for self-similar processes

J. M. P. Albin

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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

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

First available in Project Euclid: 31 May 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Extremes Lamperti's transformation self-similar process sojourns


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

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