  Angular anisotropy techniques for cosmic diffuse radiation maps are powerful
probes, even for quite small data sets. A popular observable is the angular
power spectrum; we present a detailed study applicable to any unbinned source
skymap S(n) from which N random, independent events are observed. Its exact
variance, which is due to the finite statistics, depends only on S(n) and N; we
also derive an unbiased estimator of the variance from the data. First-order
effects agree with previous analytic estimates. Importantly, heretofore
unidentified higher-order effects are found to contribute to the variance and
may cause the uncertainty to be significantly larger than previous analytic
estimates---potentially orders of magnitude larger. Neglect of these
higher-order terms, when significant, may result in a spurious detection of the
power spectrum. On the other hand, this would indicate the presence of
higher-order spatial correlations, such as a large bispectrum, providing new
clues about the sources. Numerical simulations are shown to support these
conclusions. Applying the formalism to an ensemble of Gaussian-distributed
skymaps, the noise-dominated part of the power spectrum uncertainty is
significantly increased at high multipoles by the new, higher-order effects.
This work is important for harmonic analyses of the distributions of diffuse
high-energy gamma-rays, neutrinos, and charged cosmic rays, as well as for
populations of sparse point sources such as active galactic nuclei.
