Let $U$ be a unitary operator acting on the Hilbert space $\ch$, and
$\a:\{1,..., 2k\}\mapsto\{1,..., k\}$ a pair partition. Then the ergodic
average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1}
U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} $$
converges in the strong operator topology provided $U$ is almost periodic, that
is when $\ch$ is generated by the eigenvalues of $U$. We apply the present
result to obtain the convergence of the Cesaro mean of several multiple
correlations.