  By a classical principle of analysis, sufficiently thin subsequences of
general sequences of functions behave like sequences of independent random
variables. This observation not only explains the remarkable properties of
lacunary trigonometric series, but also provides a powerful tool in many areas
of analysis. In contrast to "true" random processes, however, the probabilistic
structure of lacunary sequences is not permutation-invariant and the analytic
properties of such sequences can change radically after rearrangement. The
purpose of this paper is to survey some recent results of the authors on
permuted function series. We will see that rearrangement properties of lacunary
trigonometric series $\sum (a_k\cos n_kx+b_k \sin n_kx)$ and their nonharmonic
analogues $\sum c_k f(n_kx)$ are intimately connected with the number theoretic
properties of $(n_k)_{k \geq 1}$ and we will give a complete characterization
of permutational invariance in terms of the Diophantine properties of $(n_k)_{k
\geq 1}$. We will also see that in a certain statistical sense, permutational
invariance is the "typical" behavior of lacunary sequences.
