### Extreme events cutoff,long-range correlation and linearisation effect in multifractal analysis,

F. Angeletti, Extreme Value Analysis, Lyon, (2011), ### Abstract

The analysis of the linearization effect in multifractal analysis [2,3],and hence of the estimation of moments for multifracta lprocesses,is revisited borrowing concepts from the statistical physics of disordered systems,notably from the analysis of the so-called Random Energy Model [4]. Considering a standard multifractal process (compound Poisson motion [5]),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.