  The power density spectrum of a light curve is often calculated as the
average of a number of spectra derived on individual time intervals the light
curve is divided into. This procedure implicitly assumes that each time
interval is a different sample function of the same stochastic ergodic process.
While this assumption can be applied to many astrophysical sources, there
remains a class of transient, highly nonstationary and short-lived events, such
as gamma-ray bursts, for which this approach is often inadequate. The power
spectrum statistics of a constant signal affected by statistical (Poisson)
noise is known to be a chi2(2) in the Leahy normalisation. However, this is no
more the case when a nonstationary signal is also present. As a consequence,
the uncertainties on the power spectrum cannot be calculated based on the
chi2(2) properties, as assumed by tools such as XRONOS powspec. We generalise
the result in the case of a nonstationary signal affected by uncorrelated white
noise and show that the new distribution is a non-central chi2(2,lambda), whose
non-central value lambda is the power spectrum of the deterministic function
describing the nonstationary signal. Finally, we test these results in the case
of synthetic curves of gamma-ray bursts. We end up with a new formula for
calculating the power spectrum uncertainties. This is crucial in the case of
nonstationary short-lived processes affected by uncorrelated statistical noise,
for which ensemble averaging does not make any physical sense.
