Invariance conditions for random curvature models (Q1417675)

The author introduces a class of probability laws suggesed by the geometric optics of the human eye. These models are concerned with the representation of random corneal curvature measurements \(y\) indexed by concentric equally-spaced locations \(v= \{\theta_1,\theta_2,\dots, \theta_\ell\}\). He considers the effect that the symmetries imposed on the probability law of \(y\) have on the probability law for the ranking permutations associated with the ordering of the observed curvatures, as well as on the resolution geometric optics (astigmatic or sitmatic) that is consistent with these symmetries. In Section 2, after the definitions of astigmatic and stigmatic probability laws, the notations and structured data formulations needed for the remaining sections are given. In Section 3, the first moments of the angular variation are derived when the law of \(y\) is permutation symmetric. The case of finite-valued curvatures is discussed in Section 4, whereas a constructive rule for obtaining astigmatic laws is introduced in Section 5. Additional comments and technical definitions are given in Section 6.