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On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes

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Abstract

With motivation from Hüsler (Extremes 7:179–190, 2004) and Piterbarg (Extremes 7:161–177, 2004) in this paper we derive the joint limiting distribution of standardised maximum of a continuous, stationary Gaussian process and the standardised maximum of this process sampled at discrete time points. We prove that these two random sequences are asymptotically complete dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the grid is sufficiently sparse. We show that our results are relevant for computational problems related to discrete time approximation of the continuous time maximum.

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Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, People’s Republic of China

    Zhongquan Tan

  2. Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, Lausanne, Switzerland

    Enkelejd Hashorva

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  1. Zhongquan Tan
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  2. Enkelejd Hashorva
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Correspondence to Enkelejd Hashorva.

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Tan, Z., Hashorva, E. On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes. Methodol Comput Appl Probab 16, 169–185 (2014). https://doi.org/10.1007/s11009-012-9305-8

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  • DOI: https://doi.org/10.1007/s11009-012-9305-8

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