Random Operator-Valued Kernels and Moment Dilations

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.