Kernel Radon-Nikodym Derivatives for Random Matrix Products
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918407987789824 |
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| author | Tian, James |
| author_facet | Tian, James |
| contents | 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 ${Ψ\left(I\right)\le I}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20496 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kernel Radon-Nikodym Derivatives for Random Matrix Products Tian, James Functional Analysis Operator Algebras Probability Primary 47A20, Secondary 46E22, 46L53, 60B20 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 ${Ψ\left(I\right)\le I}$. |
| title | Kernel Radon-Nikodym Derivatives for Random Matrix Products |
| topic | Functional Analysis Operator Algebras Probability Primary 47A20, Secondary 46E22, 46L53, 60B20 |
| url | https://arxiv.org/abs/2509.20496 |