  By a classical principle of probability theory, sufficiently thin
subsequences of general sequences of random variables behave like i.i.d.\
sequences. This observation not only explains the remarkable properties of
lacunary trigonometric series, but also provides a powerful tool in many areas
of analysis, such the theory of orthogonal series and Banach space theory. In
contrast to i.i.d.\ sequences, however, the probabilistic structure of lacunary
sequences is not permutation-invariant and the analytic properties of such
sequences can change after rearrangement. In a previous paper we showed that
permutation-invariance of subsequences of the trigonometric system and related
function systems is connected with Diophantine properties of the index
sequence. In this paper we will study permutation-invariance of subsequences of
general r.v.\ sequences.
