A Note on Sums of Independent Random Variables

provided (Xn) are either symmetric or positive, and in the first case p ≥ 2, and in the second case p ≥ 1. The main novelty here is the fact that, contrary to the classical inequalities, the constants here are independent of p. Certain particular cases of Lata la’s result had been known earlier (see e.g. Hitczenko (1993), Gluskin and Kwapień (1995) or Hitczenko, Montgomery-Smith and Oleszkiewicz (1997)), but they can be easily deduced from Lata la’s inequality. Of course, the ultimate goal is to obtain bounds on the tail probabilities for sums of random variables. Lata la’s result prompted us to investigate that problem. This program has been completed; our methods, which are based on estimates for the decreasing rearrangement of a random variable, work in a rather general setting. As a result we were able to obtain extensions of Lata la’s result in various directions. The details of that approach will be presented elsewhere. The goal of this note is quite different; we will present a very simple argument that allows