Sums of independent and freely independent identically distributed random variables

Let E be a symmetric (quasi-)Banach function space on (0, 1): It is proved that every sequence of independent identically and symmetrically distributed random variables in E spans l(2) provided that E is an interpolation space for the couple (L-2, exp(L-2)): We prove that the Khinchin inequality holds in E for arbitrary independent mean zero random variables if and only if it holds for arbitrary independent identically distributed mean zero random variables. Our results complement and strengthen earlier results of Braverman and Astashkin. We also consider noncommutative analogues for freely independent random variables. The latter case demonstrates substantially better behavior than the commutative case.