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Kernel Radon-Nikodym Derivatives for Random Matrix Products

James Tian

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 $Ψ$, and the shifted kernel inequality is equivalent to the condition ${Ψ\left(I\right)\le I}$.

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 , which yields an explicit coordinate formula for the fibers. In the Gaussian Kraus model, the diagonal moments are generated by a completely positive map , and the shifted kernel inequality is equivalent to the condition .

Paper Structure

This paper contains 4 sections, 12 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gaussian products with inhomogeneous variances
  4. Gaussian Kraus increments

Key Result

Proposition 2.1

Let $X$ be a set, and let $K,L:X\times X\to\mathcal{L}\left(H\right)$ be positive definite kernels with $K\le L$. Let be a minimal Kolmogorov decomposition of $L$ on $\mathcal{K}_{L}$. Then there exists a unique operator $D\in\mathcal{L}\left(\mathcal{K}_{L}\right)$ such that

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 17 more