We find a general formula for the distribution of time-averaged observables
for systems modeled according to the sub-diffusive continuous time random walk.
For Gaussian random walks coupled to a thermal bath we recover ergodicity and
Boltzmann's statistics, while for the anomalous subdiffusive case a weakly
non-ergodic statistical mechanical framework is constructed, which is based on
L\'evy's generalized central limit theorem. As an example we calculate the
distribution of Xˉ: the time average of the position of the particle,
for unbiased and uniformly biased particles, and show that Xˉ exhibits
large fluctuations compared with the ensemble average .Comment: 5 pages, 2 figure