Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors

Let X-i:(Omega, P) --> R-n be an i.i.d. sequence of rotationally invariant random vectors in R-n. If parallel toX(1)parallel to (2) is dominated (in t he sense defined below) by parallel toZ parallel to (2) for a rotationally invariant normal random vector Z in R-n, then for each k is an element ofN and (alpha (i)) subset of or equal to R [GRAPHICS] for p greater than or equal to3 or p,n greater than or equal to2 (resp. for 1 less than or equal top less than or equal to2, n greater than or equal t o3). The constant (E parallel toZ parallel to (p))(1/p) is the best possibl e. The result applies, in particular, for variables uniformly distributed o n the sphere Sn-1 or the ball B-n. In the case of the sphere, the best cons tant is (E parallel toZ parallel to (p))(1/p) = root2/n (Gamma (p+n/2)\Gamma (n/2)) (1/p). With this constant, the Khintchine type inequality in this case also holds for 1 less than or equal top less than or equal to2,n=2.