  For the partial sums formed from a sequence of i.i.d. random variables having
a finite absolute p'th moment for some p in (0,2), we extend the recent and
striking discovery of Hechner and Heinkel (Journal of Theoretical Probability
(2010)) concerning "complete moment convergence" to the two cases 0<p<1 and
p=1. Moreover, for 0<p<2, we obtain "almost sure convergence" analogues of
these "complete moment convergence" results and these "almost sure convergence"
analogues may be regarded as being a refinement of the celebrated
Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. Versions of the
above results in a Banach space setting are also presented.
