  We study time averages of single particle trajectories in scale free
anomalous diffusion processes, in which the measurement starts at some time
t_a>0 after initiation of the process at the time origin, t=0. Using ageing
renewal theory we show that for such non-stationary processes a large class of
observables are affected by a unique ageing function, which is independent of
boundary conditions or the external forces. We quantify the weakly non-ergodic
nature of this process in terms of the distribution of time averages and the
ergodicity breaking parameter which both explicitly depend on the ageing time
t_a. Consequences for the interpretation of single particle tracking data are
discussed.
