We discuss methods currently in use for determining the significance of peaks in the periodograms of time series. We discuss some general methods for constructing significance tests, false alarm probability functions, and the role played in these by independent random variables and by empirical and theoretical cumulative distribution functions. We also discuss the concept of independent frequencies in periodogram analysis. We propose a practical method for estimating the significance of periodogram peaks, applicable to all time series irrespective of the spacing of the data. This method, based on Monte Carlo simulations, produces significance tests that are tailor-made for any given astronomical time series.
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