  The term "false-alarm probability" denotes the probability that at least one
out of M independent power values in a prescribed search band of a power
spectrum computed from a white-noise time series is expected to be as large as
or larger than a given value. The usual formula is based on the assumption that
powers are distributed exponentially, as one expects for power measurements of
normally distributed random noise. However, in practice one typically examines
peaks in an over-sampled power spectrum. It is therefore more appropriate to
compare the strength of a particular peak with the distribution of peaks in
over-sampled power spectra derived from normally distributed random noise. We
show that this leads to a formula for the false-alarm probability that is more
conservative than the familiar formula. We also show how to combine these
results with a Bayesian method for estimating the probability of the null
hypothesis (that there is no oscillation in the time series), and we discuss as
an example the application of these procedures to Super-Kamiokande solar
neutrino data.
