  Using fits to numerical simulations, we show that the entire hierarchy of
moments quickly ceases to provide a complete description of the convergence
one-point probability density function leaving the linear regime. This suggests
that the full N-point correlation function hierarchy of the convergence field
becomes quickly generically incomplete and a very poor cosmological probe on
nonlinear scales. At the scale of unit variance, only 5% of the Fisher
information content of the one-point probability density function is still
contained in its hierarchy of moments, making clear that information escaping
the hierarchy is a far stronger effect than information propagating to higher
order moments. It follows that the constraints on cosmological parameters
achievable through extraction of the entire hierarchy become suboptimal by
large amounts. A simple logarithmic mapping makes the moment hierarchy well
suited again for parameter extraction.
