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We trace with analytical methods and in a model parameter independent mannerthe independent bits of Fisher information of each of the moments of thelognormal distribution, as a now standard prescription for the distribution ofthe cosmological matter density field, as it departs from Gaussian initialconditions. We show that, when entering the regime of large fluctuations, onlya tiny, dramatically decaying fraction of the total information content remainsaccessible through the extraction of the full series of moments of the field.This is due to a known peculiarity of highly tailed distributions, that theycannot be uniquely recovered given the values of all of their moments. Thisrenders under this lognormal assumption cosmological probes such as thecorrelation function hierarchy or equivalently their Fourier transformsfundamentally limited once the field becomes non linear, for any parameter ofinterest. We show that the fraction of the information accessible fromtwo-point correlations decays to zero following the inverse squared variance ofthe field. We discuss what general properties of a random field's probabilitydensity function are making the correlation function hierarchy an efficient orinefficient, complete or incomplete set of probes of any model parameter.

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