Review on the Limiting Behavior of Tail Series of Independent Summands

For the almost certainly convergent series $S_n$ of independent random variables the limiting behavior of tail series ${T_n}{\equiv}S-S_{n-1}$ is reviewed. More specifically, tail series strong laws of large number and tail series weak laws of large numbers will be introduced, and their relationship will be investigated. Then, the relationship will also be extended to the case of Banach space valued random elements, by investigating the duality between the limiting behavior of the tail series of random variables and that of random elements.