  Stationary processes do not accurately describe the diffuse backgrounds of
relic gravitons whose correlations are homogeneous in space (i.e. only
dependent upon the distance between the two spatial locations) but not in time.
The symmetries of the autocorrelations ultimately reflect the quantum
mechanical origin of the diffuse backgrounds and lead to non-stationary
observables at late time. In particular, large oscillations are believed to
arise in the spectral energy density that is customarily (but approximately)
related to the tensor power spectrum. When the full expression of the spectral
energy density is employed the amplitudes of oscillation are instead suppressed
in the large-scale limit and the non-stationary features of the late-time
signal practically disappear. For similar reasons the relations between the
spectral energy density and the spectral amplitude are ambiguous in the
presence of non-stationary features. While it is debatable if the
non-stationary features are (or will be) directly detectable, we argue that the
spectral amplitude following from the Wiener-Khintchine theorem is generally
inappropriate for a consistent description of the relic signal. Nevertheless
the strong oscillatory behaviour of the late-time observables is naturally
smeared out provided the spectral energy density is selected as pivotal
variable.
