Discretization error for the maximum of a Gaussian field

Abstract : A Gaussian field $X$ defined on a square $S$ of $\mathds R^2$ is considered. We assume that this field is only observed at some points of a regular grid with spacing $\frac{1}{n}$. We are interested in the normalized discretization error $n^2(M - M_n)$, with $M$ the global maximum of $X$ over $S$ and $M_n$ the maximum of $X$ over the observation grid. The density of the location of the maximum is given using Rice formulas and its regularity is studied. Joint densities with the value of the field and the value of the second derivative are also given. Then, a kind of Slepian model is used to study the field behavior around the unique point where the maximum is attained, called $t^*$. We show that the normalized discretization error can be bounded by a quantity that converges in distribution to a uniform variable. The set where this uniform variable lies principally depends on the second derivative of the field at $t^*$. The bound is a function of this quantity which is approached by finite differences in practice. The bound is applied both on simulated and real data. Real data are used in positioning by satellite systems quality assessment.
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Jean-Marc Azais, Malika Chassan. Discretization error for the maximum of a Gaussian field. Stochastic Processes and their Applications, Elsevier, 2019, ⟨10.1016/j.spa.2019.02.002⟩. ⟨hal-01513724v2⟩

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