Random Elements in Separable Hilbert Spaces

This chapter deals with random elements that take on values in a separable Hilbert space. It starts with basic facts on Hilbert spaces and operators on such spaces. Basic notions for random elements that take on values in a Hilbert space are the expectation, which is seen to be a Bochner integral, and the covariance operator, which generalizes the notion of a covariance matrix for random vectors. Under certain conditions, mean square continuous second-order stochastic processes are Hilbert space-valued random elements. In this case, the expected value and the covariance operator are given by the expected value function and the covariance function, respectively. Further topics of this chapter are the characteristic functional, the normal distribution on a separable Hilbert space, and a criterion for convergence in distribution for Hilbert space-valued random elements. A main result is a central limit theorem for triangular arrays of such random elements. The chapter proceeds with a section on weighted \(L^2\) -statistics, which are widely used for testing goodness-of-fit, and it concludes with a special collection of those statistics in the context of testing of multivariate normality.