On convergence for series of random elements

In this paper, we extend some relevant concepts for Banach space valued ran dom elements and some preliminary results are presented which are used to o btain our main results. We show that the sufficiency half of Kolmogorov's t hree series theorem holds for the convergence of series of independent rand om elements under suitable geometric conditions on the Banach space. Howeve r, the necessity part is not true in general as will be shown by a counter- example. From the main result, a strong law of numbers for independent rand om elements can be obtained. Some equivalent conditions for Banach spaces b eing of Rademacher-type p are also provided.