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Borovkov, Konstantin; Mishura, Yuliya; Novikov, Alexander; Zhitlukhin, Mikhail

Bounds for expected maxima of Gaussian processes and their discrete approximations. (English) Zbl 1361.60027

Stochastics 89, No. 1, 21-37 (2017).
Summary: The paper deals with the expected maxima of continuous Gaussian processes \(X=(X_t)_{t\geq 0}\) that are Hölder continuous in \(L_2\)-norm and/or satisfy the opposite inequality for the \(L_2\)-norms of their increments. Examples of such processes include the fractional Brownian motion and some of its “relatives” (of which several examples are given in the paper). We establish upper and lower bounds for \(E\) \(\max_{0\leq t\leq 1} X_t\) and investigate the rate of convergence to that quantity of its discrete approximation \(E\) \(\max_{0\leq i\leq n} X_{i/n}\). Some further properties of these two maxima are established in the special case of the fractional Brownian motion.

Cited in 20 Documents

MSC:

60G15 Gaussian processes
60G70 Extreme value theory; extremal stochastic processes
60G22 Fractional processes, including fractional Brownian motion
60J65 Brownian motion

Keywords:

Gaussian processes; expected maximum; fractional Brownian motion; discrete approximation; long memory processes
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