Extreme values and fat tails of multifractal fluctuations.

In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, the standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non-self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framework that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed "inverse cubic law" of the return pdf tails.

Multifractal random walk.

We introduce a class of multifractal processes, referred to as multifractal random walks (MRWs). To our knowledge, it is the first multifractal process with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.

Long-range correlations in genomic DNA: a signature of the nucleosomal structure.

We use the "wavelet transform microscope" to carry out a comparative statistical analysis of DNA bending profiles and of the corresponding DNA texts. In the three kingdoms, one reveals on both signals a characteristic scale of 100-200 bp that separates two different regimes of power-law correlations (PLC). In the small-scale regime, PLC are observed in eukaryotic, in double-strand DNA viral, and in archaeal genomes, which contrasts with their total absence in the genomes of eubacteria and their viruses. This strongly suggests that small-scale PLC are related to the mechanisms underlying the wrapping of DNA in the nucleosomal structure. We further speculate that the large scale PLC are the signature of the higher-order structure and dynamics of chromatin.