  In this paper, motived by the notion of independent and identically
distributed random variables under the sub-linear expectation initiated by
Peng, we give a theorem about the convergence of a random series and establish
a three series theorem of independent random variables under the sub-linear
expectations. As an application, we obtain the Marcinkiewicz's strong law of
large numbers for independent and identically distributed random variables
under the sub-linear expectations. The technical details are different from
those for classical theorems because the sub-linear expectation and its related
capacity are not additive.
