Let 1\leq p<2 and let L_p=L_p[0,1] be the classical L_p-space of all (classes
of) p-integrable functions on [0,1]. It is known that a sequence of independent
copies of a mean zero random variable f from L_p spans in L_p a subspace
isomorphic to some Orlicz sequence space l_M. We present precise connections
between M and f and establish conditions under which the distribution of a
random variable f whose independent copies span l_M in L_p is essentially
unique.Comment: 14 pages, submitte