Schoenberg’s Theorem and Unitarily Invariant Random Arrays

Motivated by Schoenberg’s theorem in classical analysis and some recent work on the circular law, we extend the basic representations of unitarily invariant random arrays and functionals to the complex case. As in the real case, the basic building blocks are multiple Wiener–Itô integrals on tensor products of a separable Hilbert space. The complex setting leads to some unexpected simplifications.