  We consider the "multi-frequency" periodogram, in which the putative signal
is modelled as a sum of two or more sinusoidal harmonics with idependent
frequencies. It is useful in the cases when the data may contain several
periodic components, especially when their interaction with each other and with
the data sampling patterns might produce misleading results.
  Although the multi-frequency statistic itself was already constructed, e.g.
by G. Foster in his CLEANest algorithm, its probabilistic properties (the
detection significance levels) are still poorly known and much of what is
deemed known is unrigourous. These detection levels are nonetheless important
for the data analysis. We argue that to prove the simultaneous existence of all
$n$ components revealed in a multi-periodic variation, it is mandatory to apply
at least $2^n-1$ significance tests, among which the most involves various
multi-frequency statistics, and only $n$ tests are single-frequency ones.
  The main result of the paper is an analytic estimation of the statistical
significance of the frequency tuples that the multi-frequency periodogram can
reveal. Using the theory of extreme values of random fields (the generalized
Rice method), we find a handy approximation to the relevant false alarm
probability. For the double-frequency periodogram this approximation is given
by an elementary formula $\frac{\pi}{16} W^2 e^{-z} z^2$, where $W$ stands for
a normalized width of the settled frequency range, and $z$ is the observed
periodogram maximum. We carried out intensive Monte Carlo simulations to show
that the practical quality of this approximation is satisfactory. A similar
analytic expression for the general multi-frequency periodogram is also given
in the paper, though with a smaller amount of numerical verification.
