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.