  Let (f_n) and (g_n) be two sequences of random variables adapted to an
increasing sequence of $\sigma$-algebras $({\cal F}_n)$ such that the
conditional distributions of f_n and g_n given ${\cal F}_{n-1}$ coincide, and
such that the sequence (g_n) is conditionally independent. Then it is known
that $\normo{\sum f_k}_p \le C \normo{\sum g_k}_p$, $1 \le p \le \infty$ where
the constant C is independent of p. The aim of this paper is to extend this
result to certain classes of Orlicz and rearrangement invariant spaces. This
paper includes fairly general techniques for obtaining rearrangement invariant
inequalities from Orlicz norm inequalities.
