Consistent Estimates of Deformed Isotropic Gaussian Random Fields on the Plane

October, 2007
Report Number: 
Ethan Anderes and Sourav Chatterjee

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation $f\colon \Bbb R^2\rightarrow \Bbb R^2$ when observing the deformed random field $Z\circ f$ on a dense grid in a bounded simply connected domain $\Omega$ where $ Z$ is assumed to be an isotropic Gaussian random field on $\Bbb R^2$. The estimate, $\hat f$, is constructed on a simply connected domain $U$ such that $\overline U\subset\Omega$ and is defined using kernel smoothed quadratic variations, Bergman projections and results from quasiconformal theory. We show under mild assumptions on the random field $Z$ and the deformation $f$ that $\hat f\rightarrow R_\theta f+c$ uniformly on compact subsets of $U$ with probability one as the grid spacing goes to zero, where $R_\theta$ is an unidentifiable rotation and $c$ is an unidentifiable translation.

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