Extremes Values of Discrete and Continuous Time Strongly Dependent Gaussian Processes

Abstract

The joint limit distribution of the maximum of a continuous, strongly dependent stationary Gaussian process and the maximum of this process sampled at discrete time points is studied. It is shown that these two extreme values are asymptotically totally dependent if the grid of the discrete time points is sufficiently dense, and asymptotically dependent if the the grid points are sparse or Pickands grids. Our results are motivated by the deep contributions Piterbarg (2004) and Hüsler (2004).

Keywords:

• Continuous time process
• Dependence
• Discrete time process
• Extreme values
• Gaussian processes

Mathematics Subject Classification:

• Primary 60F05
• Secondary 60G15

Acknowledgments

The authors would like to thank the referees and the Associate Editor for their careful reading and helpful comments that improved the quality of this article.

Research supported by National Science Foundation of China (No. 11071182) and the innovation Program for graduate student in Jiangsu Higher Education Institutions (No. CXZZ11-0083).