  We examine the non-ergodic properties of scaled Brownian motion, a
non-stationary stochastic process with a time dependent diffusivity of the form
$D(t)\simeq t^{\alpha-1}$. We compute the ergodicity breaking parameter EB in
the entire range of scaling exponents $\alpha$, both analytically and via
extensive computer simulations of the stochastic Langevin equation. We
demonstrate that in the limit of long trajectory lengths $T$ and short lag
times $\Delta$ the EB parameter as function of the scaling exponent $\alpha$
has no divergence at $\alpha=1/2$ and present the asymptotes for EB in
different limits. We generalise the analytical and simulations results for the
time averaged and ergodic properties of scaled Brownian motion in the presence
of ageing, that is, when the observation of the system starts only a finite
time span after its initiation. The approach developed here for the calculation
of the higher time averaged moments of the particle displacement can be applied
to derive the ergodic properties of other stochastic processes such as
fractional Brownian motion.
