Bounds for expected maxima of Gaussian processes and their discrete approximations

Abstract

The paper deals with the expected maxima of continuous Gaussian processes that are Hölder continuous in -norm and/or satisfy the opposite inequality for the -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 and investigate the rate of convergence to that quantity of its discrete approximation . Some further properties of these two maxima are established in the special case of the fractional Brownian motion.

Keywords:

• Expected maximum
• Gaussian processes
• fractional Brownian motion
• discrete approximation
• processes with long memory

AMS Subject Classifications:

• 60G15
• 60G22
• 60J65

Acknowledgements

The last author is grateful to V.I. Piterbarg for interesting discussions of the topic of the paper. We are grateful to the anonymous referee for her/his useful remarks that helped to improve the paper.

Notes

No potential conflict of interest was reported by the authors.

Dedicated to Bernt Øksendal on occasion of his 70th birthday.

Additional information

Funding

The work of K. Borovkov (on Sections 1 and 2) was supported by the ARC [grant number DP150102758]; A. Novikov and M. Zhitlukhin (on Sections 3 and 4) was supported by the Russian Science Foundation [grant number 14-21-00162].