  In connection with the results of Tim Austin, and Wen Huang, Song Shao,
Xiangdong Ye we present the following assertion: there are infinite
automorphisms $S,T$, some set $A$ of positive finite measure and a sequence
$N_m$ with $|N_{m+1}|/|N_m|\ \to\infty$ such that $T^iA\cap S^iA=\phi$ for
$i\in [N_{4k}, N_{4k+1}]$ and $T^iA=S^iA$ for $i\in [N_{4k+2}, N_{2k+3}]$. For
the corresponding deterministic Gaussian and Poisson suspensions
$S_\circ,T_\circ$ over $S,T$ for some $f\in L_\infty$ there is no limit of $
\frac 1 N \sum_{n=1}^N T_\circ^nf\,S_\circ^nf$ in $ L_2$.
