  For stationary signals in time the weak law of large numbers (WLLN) states
that ensemble and realization averages are within e of each other with a
probability of order O(1/Ne^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 this two
node-varying signals are within e of each other with probability of order
O(1/Ne^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 of 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.
