On the incompleteness of the moment and correlation function hierarchy as probes of the lognormal field

We trace with analytical methods and in a model parameter independent manner the independent bits of Fisher information of each of the moments of the lognormal distribution, as a now standard prescription for the distribution of the cosmological matter density field, as it departs from Gaussian initial conditions. We show that, when entering the regime of large fluctuations, only a tiny, dramatically decaying fraction of the total information content remains accessible through the extraction of the full series of moments of the field. This is due to a known peculiarity of highly tailed distributions, that they cannot be uniquely recovered given the values of all of their moments. This renders under this lognormal assumption cosmological probes such as the correlation function hierarchy or equivalently their Fourier transforms fundamentally limited once the field becomes non linear, for any parameter of interest. We show that the fraction of the information accessible from two-point correlations decays to zero following the inverse squared variance of the field. We discuss what general properties of a random field's probability density function are making the correlation function hierarchy an efficient or inefficient, complete or incomplete set of probes of any model parameter.