Ergodicity in Stationary Graph Processes: A Weak Law of Large Numbers

For stationary signals in time, the weak law of large numbers (WLLN) states that ensemble and realization averages are within epsilon of each other with a probability of order O(1/N epsilon(2)) when considering N signal components. The graph WLLN introduced in this paper shows that the same is essentially true for signals supported on graphs. However, the notions of stationarity, ensemble mean, and realization mean are different. Recent papers have defined graph stationary signals as those that satisfy a form of invariance with respect to graph diffusion. The ensemble mean of a graph stationary signal is not a constant but a node-varying signal whose structure depends on the spectral properties of the graph. The realization average of a graph signal is defined here as an average of successive weighted averages of local signal values with signal values of neighboring nodes. The graph WLLN shows that these two node-varying signals are within epsilon of each other with probability of order O(1/N epsilon(2)) in at least some nodes. In stationary time signals, the realization average is not only a consistent estimator of the ensemble mean but also optimal in terms of mean squared error (MSE). This is not true for graph signals. Optimal MSE graph filter designs are also presented. An example problem concerning the estimation of the mean of a Gaussian random field is presented.