  We derive fundamental limitations on the performances of intrinsic averaging
algorithms in open multi-agent systems, which are systems subject to random
arrivals and departures of agents. Each agent holds a value, and their goal is
to estimate the average of the values of the agents presently in the system. We
provide a lower bound on the expected Mean Square Error for any estimation
algorithm, assuming that the number of agents remains constant and that
communications are random and pairwise. Our derivation is based on the expected
error obtained with an optimal algorithm under conditions more favorable than
those the actual problem allows, and relies on an analysis of the constraints
on the information spreading mechanisms in the system, and relaxations of
these.
