Strong laws for sums of extreme values

We present a new approach to strong laws for sums of extreme values of sequences of independent not necessarily identically distributed random variables. Our technique is based on a simple extension of an inequality of M. Marcus and G. Pisier [M. Marcus and G. Pisier, Acta Math., 152 (1984), pp. 245–301.] on the tail distribution of weak $l_p$ norms of independent sequences of real random variables. Our results are also related to some extensions of the Marcinkiewicz–Zygmund law of large numbers.