  In systems with weak ergodicity breaking, the equivalence of time averages
and ensemble averages is known to be broken. We study here the computation of
the power spectrum from realizations of a specific process exhibiting 1/f^beta
noise, the Rebenshtok-Barkai model. We show that even the binned power spectrum
does not converge in the limit of infinite time, but that instead the resulting
value is a random variable stemming from a distribution with finite variance.
However, due to the strong correlations in neighboring frequency bins of the
spectrum, the exponent beta can be safely estimated by time averages of this
type. Analytical calculations are illustrated by numerical simulations.
