A Weak Convergence Theorem for Random Sums of Independent Random Variables

Abstract

A limit theorem for random sums of independent random elements of $D\lbrack 0, 1\rbrack$ is proved. The theorem extends and simplifies the proof of a result obtained by R. Pyke in [3]. In a personal communication Professor Pyke remarked to me that the method employed is also the right approach to the problem studied in [4], by adequately defining sequences of random elements of $D\lbrack 0, 1 \rbrack$.