Characterization of weak limits of randomly indexed sequences

Let {Xn, n[greater-or-equal, slanted]1} be an arbitrary sequence of random elements defined on a probability space with a nonatomic measure P and taking values in a separable complete metric space (S,[rho]). In this paper we characterize the set of all possible weak limits of the sequences {XNn, n[greater-or-equal, slanted]1}, where {Nn, n[greater-or-equal, slanted]1} is a sequence of positive integer-valued random variables. The proof shows how, for a given probability law F( ), we can define a random sequence {Nn, n[greater-or-equal, slanted]1} satisfying .