Séminaires DSA -Discretization error for the maximum of a Gaussian field

A Gaussian field $X$ defined on a square $T$ of $\R^2$ is considered. We assume that this field is only observed at some points of a regular grid with spacing $ rac{1}{n}$. We are interested in the discretization error $M - M_n$, with $M$ the global maximum of $X$ over $T$ and $M_n$ the maximum of $X$ over the observation grid. Using a model inspired by Slepian models, an asymptotic equivalent of the discretization error is given and thus an asymptotic bound for this error.