  The analysis of the linearization effect in multifractal analysis, and hence
of the estimation of moments for multifractal processes, is revisited borrowing
concepts from the statistical physics of disordered systems, notably from the
analysis of the so-called Random Energy Model. Considering a standard
multifractal process (compound Poisson motion), chosen as a simple
representative example, we show: i) the existence of a critical order $q^*$
beyond which moments, though finite, cannot be estimated through empirical
averages, irrespective of the sample size of the observation; ii) that
multifractal exponents necessarily behave linearly in $q$, for $q > q^*$.
Tayloring the analysis conducted for the Random Energy Model to that of
Compound Poisson motion, we provide explicative and quantitative predictions
for the values of $q^*$ and for the slope controlling the linear behavior of
the multifractal exponents. These quantities are shown to be related only to
the definition of the multifractal process and not to depend on the sample size
of the observation. Monte-Carlo simulations, conducted over a large number of
large sample size realizations of compound Poisson motion, comfort and extend
these analyses.
