  The cross-spectrum method consists in measuring a signal $c(t)$
simultaneously with two independent instruments. Each of these instruments
contributes to the global noise by its intrinsec (white) noise, whereas the
signal $c(t)$ that we want to characterize could be a (red) noise. We first
define the real part of the cross-spectrum as a relevant estimator. Then, we
characterize the probability density function (PDF) of this estimator knowing
the noise level (direct problem) as a Variance-Gamma (V$\Gamma$) distribution.
Next, we solve the "inverse problem" thanks to Bayes' theorem to obtain an
upper limit of the noise level knowing the estimate. Checked by massive Monte
Carlo simulations, V$\Gamma$ proves to be perfectly reliable to any number of
degrees of freedom (dof). Finally we compare this method with an other method
using the Karhunen-Lo\`{e}ve transfrom (KLT). We find an upper limit of the
signal level slightly different as the one of V$\Gamma$ since KLT better takes
into account the available informations.
