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Motivated by recent results on lognormal statistics showing that the momenthierarchy of a lognormal variable completely fails at capturing its informationcontent in the large variance regime, we discuss in this work the inadequacy ofthe hierarchy of correlation functions to describe a correlated lognormalfield, which provides a roughly accurate description of the non-linearcosmological matter density field. We present families of fields having thesame hierarchy of correlation functions than the lognormal field at all orders.This explicitly demonstrates the little studied though known fact that thecorrelation function hierarchy never provides a complete description of alognormal field, and that it fails to capture information in the non-linearregime, where other simple observables are left totally unconstrained. Wediscuss why perturbative, Edgeworth-like approaches to statistics in thenon-linear regime, common in cosmology, can never reproduce or predict thateffect, and why it is however generic for tailed fields, hinting at a breakdownof the perturbation theory based on the field fluctuations. We make a rough butsuccessful quantitative connection to N-body simulations results, that showedthat the spectrum of the log-density field carries more information than thespectrum of the field entering the non-linear regime.

On the inadequacy of N-point correlation functions to describe nonlinear cosmological fields: explicit examples and connection to simulations