  A cyclic urn is an urn model for balls of types $0,\ldots,m-1$ where in each
draw the ball drawn, say of type $j$, is returned to the urn together with a
new ball of type $j+1 \mod m$. The case $m=2$ is the well-known Friedman urn.
The composition vector, i.e., the vector of the numbers of balls of each type
after $n$ steps is, after normalization, known to be asymptotically normal for
$2\le m\le 6$. For $m\ge 7$ the normalized composition vector does not
converge. However, there is an almost sure approximation by a periodic random
vector. In this paper the asymptotic fluctuations around this periodic random
vector are identified. We show that these fluctuations are asymptotically
normal for all $m\ge 7$. However, they are of maximal dimension $m-1$ only when
$6$ does not divide $m$. For $m$ being a multiple of $6$ the fluctuations are
supported by a two-dimensional subspace.
