General and consistent statistics for cosmological observations
  This paper focuses on two aspects of the statistics of cosmological
observables that are important for the next stages of precision cosmology.
First, we note that the theory of reduced angular $N$-point spectra has only
been developed in detail up to the trispectrum case and in a fashion that makes
it difficult to go beyond. To fill this gap, here we present a constructive
approach that provides a systematic description of reduced angular $N$-point
spectra and their covariance matrices, for arbitrary $N$. Second, we focus on
the common practice in the literature on cosmological observables, which
consists in simply discarding a part of the expression, namely, the terms
containing fields evaluated at the observer position. We point out that this is
not justified beyond linear order in perturbation theory, as these terms
contribute to all the multipoles of the corresponding spectra and with a
magnitude that is of the same order as the rest of the non-linear corrections.
We consider the possibility that the reason for neglecting these terms is a
conceptual discomfort when using ensemble averages, which originates in an
apparent tension between the ergodic hypothesis and the privileged position of
the observer on the light-cone. We clarify this subtle issue by performing a
careful derivation of the relation between the theoretical statistical
predictions and the observational estimators for all $N$. We conclude that
there is no inconsistency whatsoever in ensemble-averaging fields at and near
the observer position, thus clearing the way for consistent and robust
high-precision calculations.
