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We consider the "multi-frequency" periodogram, in which the putative signalis modelled as a sum of two or more sinusoidal harmonics with idependentfrequencies. It is useful in the cases when the data may contain severalperiodic components, especially when their interaction with each other and withthe 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 (thedetection significance levels) are still poorly known and much of what isdeemed known is unrigourous. These detection levels are nonetheless importantfor 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 applyat least $2^n-1$ significance tests, among which the most involves variousmulti-frequency statistics, and only $n$ tests are single-frequency ones.  The main result of the paper is an analytic estimation of the statisticalsignificance of the frequency tuples that the multi-frequency periodogram canreveal. Using the theory of extreme values of random fields (the generalizedRice method), we find a handy approximation to the relevant false alarmprobability. For the double-frequency periodogram this approximation is givenby an elementary formula $\frac{\pi}{16} W^2 e^{-z} z^2$, where $W$ stands fora normalized width of the settled frequency range, and $z$ is the observedperiodogram maximum. We carried out intensive Monte Carlo simulations to showthat the practical quality of this approximation is satisfactory. A similaranalytic expression for the general multi-frequency periodogram is also givenin the paper, though with a smaller amount of numerical verification.

Detecting multiple periodicities in observational data with the multi-frequency periodogram. I. Analytic assessment of the statistical significance