Random Operator-Valued Kernels and Moment Dilations

TL;DR

This paper introduces and analyzes random operator-valued positive definite kernels, establishing their properties, connections to moment dilations, and implications for operator-theoretic realizations in a probabilistic setting.

Contribution

It extends deterministic operator-valued kernels to a random setting with expectation-based positivity and develops a moment dilation framework linking statistical moments to operator dilations.

Findings

Established convexity and sampling stability of random kernels

Proved Gaussian factorization under Hilbert-Schmidt/trace conditions

Linked moment dilation to unitary dynamics and operator realization

Abstract

This paper studies random operator-valued positive definite (p.d.) kernels and their connection to moment dilations. A class of random p.d. kernels is introduced in which the positivity requirement is imposed only in expectation, extending the deterministic case. Basic properties are established, including convexity, sampling stability, and the relationship between mean kernels and deterministic ones. A Gaussian factorization result is proved under a Hilbert-Schmidt/trace condition, clarifying the difference between pathwise and mean-square positive definiteness and linking the construction to radonification in the non-trace-class setting. Moment dilation is then formulated for random operators, extending classical dilation theory to preserve equality of mixed moments in expectation. The resulting dilation triples admit interpretation in terms of unitary dynamics, connecting statistical moment structure with operator-theoretic realization.