The Probability Distribution for Non-Gaussianity Estimators
  One of the principle efforts in cosmic microwave background (CMB) research is
measurement of the parameter fnl that quantifies the departure from Gaussianity
in a large class of non-minimal inflationary (and other) models. Estimators for
fnl are composed of a sum of products of the temperatures in three different
pixels in the CMB map. Since the number ~Npix^2 of terms in this sum exceeds
the number Npix of measurements, these ~Npix^2 terms cannot be statistically
independent. Therefore, the central-limit theorem does not necessarily apply,
and the probability distribution function (PDF) for the fnl estimator does not
necessarily approach a Gaussian distribution for N_pix >> 1. Although the
variance of the estimators is known, the significance of a measurement of fnl
depends on knowledge of the full shape of its PDF. Here we use Monte Carlo
realizations of CMB maps to determine the PDF for two minimum-variance
estimators: the standard estimator, constructed under the null hypothesis
(fnl=0), and an improved estimator with a smaller variance for |fnl| > 0. While
the PDF for the null-hypothesis estimator is very nearly Gaussian when the true
value of fnl is zero, the PDF becomes significantly non-Gaussian when |fnl| >
0. In this case we find that the PDF for the null-hypothesis estimator fnl_hat
is skewed, with a long non-Gaussian tail at fnl_hat > |fnl| and less
probability at fnl_hat < |fnl| than in the Gaussian case. We provide an
analytic fit to these PDFs. On the other hand, we find that the PDF for the
improved estimator is nearly Gaussian for observationally allowed values of
fnl. We discuss briefly the implications for trispectrum (and other
higher-order correlation) estimators.
