Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities

Abstract

The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the p p moment of a sum of independent symmetric random variables to that of the p p and 2 2 moments of the individual variables, are computed in the range 2 > p ≤ 4 2>p\le 4 . This complements the work of Utev who has done the same for p > 4 p>4 . The qualitative nature of the extreme cases turns out to be different for p > 4 p>4 than for p > 4 p>4 . The method developed yields results in some more general and other related moment inequalities.