Schoenberg's theorem and unitarily invariant random arrays
A classical theorem of Schoenberg states, in one of possible formulations, that, if \(\xi_1,\xi_2,\dots\) is a sequence of real random variables, then the distribution of \((\xi_1,\dotsc,\xi_n)\) is rotationally invariant for every \(n\in \mathbb N\) if and only if the \(\xi_n\) have a representation of the form \(\xi_i=\sigma N_i\), where \(N_i\) are i.i.d.\ standard Gaussian random variables and \(\sigma\) is a random variable independent of the \(N_i\). In an earlier paper [Probab. Theory Relat. Fields 103, No. 1, 91--141 (1995; Zbl 0843.60009)], \textit{O. Kallenberg} extended this theorem to \(\mathbb N^d\)-indexed arrays of real random variables. In the paper under review, the author considers the complex case. Let \(H\) be a separable infinite dimensional complex Hilbert space. A continuous linear random functional (CLRF) on \(H\) is defined as a \(\mathbb C\)-valued stochastic process \(\{X(h): h\in H\}\) such that (1) \(X(a_1h_1+a_2h_2) = a_1 X(h_1) +a_2 X(h_2)\) a.s.\ for all \(h_1,h_2\in H\) and \(a_1,a_2\in\mathbb C\), (2) \(X(h_n)\) tends in probability to \(0\) for every sequence \(\{h_n\}_{n\in\mathbb N}\) in \(H\) such that \(\|h_n\|\) tends to \(0\). Let now \(X\) be a CLRF on \(H^{\otimes d}\). One says that \(X\) is jointly rotatable if \(X\circ U\) equals in distribution to \(X\) for every unitary operator \(U\) on \(H^{\otimes d}\). Further, one says that \(X\) is separately rotatable if \(X\circ \otimes_{k=1}^d U_k\) equals in distribution to \(X\) for every unitary operators \(U_1,\dotsc,U_d\) on \(H\). The author provides a characterization of jointly and separately rotatable CLRFs. A similar characterization in the setting of real Hilbert spaces can be found in the paper cited above. The basic building blocks of this characterization are the multiple Wiener-Itō integrals. Let \(e_1,e_2,\dots\) be an orthonormal basis in \(H\). To a CLRF \(X\) on \(H^{\otimes d}\), one can associate a complex random \(\mathbb N^{d}\)-indexed array \(X_{k_1,\dotsc,k_d}:= X(e_{k_1}\otimes \dotsb\otimes e_{k_d})\), where \((k_1,\dotsc,k_d)\in \mathbb N^d\). This makes it possible to translate the results to the setting of \(\mathbb N^d\)-indexed arrays of random variables.
scientific article; zbMATH DE number 6124633
scientific article