Discrete representations of second order random functions. I

We study representations of both scalar and vector random functions defined on a set $T$ in the form of either infinite or finite sums whose terms are scalar functions on $T$ with random coefficients. The assumptions imposed on the set $T$ and on properties of random functions are rather general. In particular, we consider the cases where $T$ is a compact topological space, measurable space with positive measure, or an arbitrary nonempty set. Various examples of such representations are given for specific random functions assuming values in Hilbert spaces. The Karhunen–Loeve type representations are considered in part I of the paper. More general basis type representations are studied in part II.