  The family of pairwise independently determined (PID) systems, i.e. those for
which the independent joining is the only self joining with independent
2-marginals, is a class of systems for which the long standing open question by
Rokhlin, of whether mixing implies mixing of all orders, has a positive answer.
We show that in the class of weakly mixing PID one finds a positive answer for
another long-standing open problem, whether the multiple ergodic averages
\begin{equation*}
  \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\cdots f_d(T^{dn}x), \quad N\to \infty,
\end{equation*} almost surely converge.
