A Refinement of the Kolmogorov-Marcinkiewicz-Zygmund Strong Law of Large Numbers

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.