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.
"Measuring the magnitude of sums of independent random variables." Ann. Probab. 29 (1) 447 - 466, February 2001. https://doi.org/10.1214/aop/1008956339