Asymptotic agreement in distributed estimation

Each of several agents updates his estimate of the same random variable whenever he makes a new observation or receives the estimate made by another agent. In turn, each agent transmits his estimate to a randomly chosen subset of the other agents. A subset of agents forms a communicating ring if for every pair <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{m}, {p}</tex> of ring members, there is a sequence of ring members <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{m} = {m}₁, {m}₂, ... , {m}ₙ₊₁ = {p}</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{m}ᵢ</tex> sends his estimate to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{m}ᵢ₊₁</tex> infinitely often. If each ring member knows that he is a ring member, then the estimates of all the ring members asymptotically agree. However, this common limit can depend upon the order in which estimates are transmitted.