  Braverman, Mallows and Shepp (1995), showed that if the absolute moments of
partial sums of i.i.d. symmetric variables are equal to those of normal
variables, then the marginals have normal distribution. This fact suggested the
conjecture that probably the absolute moments alone characterize the
homogeneous process with independent increments. In this paper we prove a more
general result that gives a positive answer to this conjecture, and then apply
it in order to obtain the CLT for a class of dependent random variables under a
normalization involving the absolute moments of partial sums.
