Measuring the magnitude of sums of independent random variables

The final version of this paper appears in: "Annals of Probability" 29 (2001): 447-466. Print.This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Lévy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms