Weak limits of consecutive projections and of greedy steps

Let $H$ be a Hilbert space. We investigate the properties of weak limit points of iterates of random projections onto $K\ge 2$ closed convex sets in $H$ and the parallel properties of weak limit points of residuals of random greedy approximation with respect to $K$ dictionaries. In the case of convex sets these properties imply weak convergence in all the cases known so far. In particular, we give a short proof of the theorem of Amemiya and Ando on weak convergence when the convex sets are subspaces. The question of the weak convergence in general remains open.