  It is a well known fact that for periodic measurable $f$ and rapidly
increasing $(n_k)_{k \geq 1}$ the sequence $(f(n_kx))_{k\ge 1}$ behaves like a
sequence of independent, identically distributed random variables. For example,
if $f$ is a periodic Lipschitz function, then $(f(2^kx))_{k\ge 1}$ satisfies
the central limit theorem, the law of the iterated logarithm and several
further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence
remains i.i.d.\ after any permutation of its terms, it is natural to expect
that the asymptotic properties of lacunary series are also
permutation-invariant. Recently, however, Fukuyama (2009) showed that a
rearrangement of the sequence $(f(2^kx))_{k\ge 1}$ can change substantially its
asymptotic behavior, a very surprising result. The purpose of the present paper
is to investigate this interesting phenomenon in detail and to give necessary
and sufficient criteria for the permutation-invariance of the CLT and LIL for
$f(n_kx)$.
