Strong law of large numbers on graphs and groups (Q2882826)

A random vertex \(x\) in a locally finite graph is assumed to have finite expected squared distance to any vertex \(v\) in the graph. The mean-set \(Ex\) of \(x\) is defined as the set of minimizers \(v\) of the expected squared distance. For a sample sequence of \(n\) i.i.d. random vertices, let \(Sn\) be the set of minimizers for the average of the \(n\) squared distances. The convergence of \(Sn\) is investigated when \(Ex\) consists of one or several vertices having positive probabilities. Limes superior of \(Sn\) is shown to be equal to \(Ex\) with probability one. Chebyshev and Chernoff bounds are also given for the probability that \(Sn\) is not included in \(Ex\).