  We study the asymptotics of the reducible representations of the wreath
products G\wr S_q=G^q \rtimes S_q for large q, where G is a fixed finite group
and S_q is the symmetric group in q elements; in particular for G=Z/2Z we
recover the hyperoctahedral groups. We decompose such a reducible
representation of G\wr S_q as a sum of irreducible components (or,
equivalently, as a collection of tuples of Young diagrams) and we ask what is
the character of a randomly chosen component (or, what are the shapes of Young
diagrams in a randomly chosen tuple). Our main result is that for a large class
of representations the fluctuations of characters (and fluctuations of the
shape of the Young diagrams) are asymptotically Gaussian. The considered class
consists of the representations for which the characters asymptotically almost
factorize and it includes, among others, the left regular representation
therefore we prove the analogue of Kerov's central limit theorem for wreath
products.
