Kernel Radon-Nikodym Derivatives for Random Matrix Products

Abstract

This paper studies kernel Radon-Nikodym derivatives for the one-step shift of time-indexed positive definite kernels associated with random matrix products. The problem is to determine when the shifted kernel is dominated by the original kernel and to identify the corresponding Radon-Nikodym derivative. We treat two concrete classes of multiplicative walks: ensembles with inhomogeneous variances and Gaussian Kraus products. In both settings, the shifted kernel inequality reduces to a one-step condition on the diagonal moments, and the Radon-Nikodym derivative is described explicitly by a fiberwise sequence in the time variable. In the inhomogeneous variance model, the diagonal compression is governed by a nonnegative matrix $S$, which yields an explicit coordinate formula for the fibers. In the Gaussian Kraus model, the diagonal moments are generated by a completely positive map $\Psi$, and the shifted kernel inequality is equivalent to the condition ${\Psi\left(I\right)\le I}$.