On Piterbarg's max-discretisation theorem for homogeneous Gaussian random fields

2015/02/04 by Zhongquan Tan, Tan, Zhongquan, Kaiyong Wang +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1502.01333

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arxiv created 2015/02/04 · openalex publication_date 2015/02/04 · arxiv updated 2015/02/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Motivated by the papers of Piterbarg (2004) and Hüsler (2004), in this paper the asymptotic relation between the maximum of a continuous dependent homogeneous Gaussian random field and the maximum of this field sampled at discrete time points is studied. It is shown that, for the weakly dependent case, these two maxima are asymptotically independent, dependent and coincide when the grid of the discrete time points is a sparse grid, Pickands grid and dense grid, respectively, while for the strongly dependent case, these two maxima 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.

Related

• Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids
• Piterbarg's max-discretisation theorem for stationary vector Gaussian processes observed on different grids
• Extremes of homogeneous Gaussian random fields
• Almost sure convergence for the maximum of nonstationary random fields
• On maximum of Gaussian random field having unique maximum point of its variance