Almost sure convergence of the general Jamison weighted sum of B-valued random variables

In this paper, two new functions are introduced to depict the Jamison weighted sum of random variables instead using the common methods, their properties and relationships are systematically discussed. We also analysed the implication of the conditions in previous papers. Then we apply these consequences to B-valued random variables, and greatly improve the original results of the strong convergence of the general Jamison weighted sum. Furthermore, our discussions are useful to the corresponding questions of real-valued random variables.