  Anomalous diffusion has been widely observed by single particle tracking
microscopy in complex systems such as biological cells. The resulting time
series are usually evaluated in terms of time averages. Often anomalous
diffusion is connected with non-ergodic behaviour. In such cases the time
averages remain random variables and hence irreproducible. Here we present a
detailed analysis of the time averaged mean squared displacement for systems
governed by anomalous diffusion, considering both unconfined and restricted
(corralled) motion. We discuss the behaviour of the time averaged mean squared
displacement for two prominent stochastic processes, namely, continuous time
random walks and fractional Brownian motion. We also study the distribution of
the time averaged mean squared displacement around its ensemble mean, and show
that this distribution preserves typical process characteristic even for short
time series. Recently, velocity correlation functions were suggested to
distinguish between these processes. We here present analytucal expressions for
the velocity correlation functions. Knowledge of the results presented here are
expected to be relevant for the correct interpretation of single particle
trajectory data in complex systems.
