Uniform convergence for complex [0,1][\mathbf{0,1}]-martingales

Positive $T$-martingales were developed as a general framework that extends the positive measure-valued martingales and are meant to model intermittent turbulence. We extend their scope by allowing the martingale to take complex values. We focus on martingales constructed on the interval $T=[0,1]$ and replace random measures by random functions. We specify a large class of such martingales for which we provide a general sufficient condition for almost sure uniform convergence to a nontrivial limit. Such a limit yields new examples of naturally generated multifractal processes that may be of use in multifractal signals modeling.Comment: Published in at http://dx.doi.org/10.1214/09-AAP664 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multiperiodic multifractal martingale measures

Projet FRACTALESA nonnegative 1-periodic multifractal measure on R is obtained as infinite random product of harmonics of a 1-periodic function W(t). Such infinite products are statistically self-affine and generalize certain Riesz products with random phases. This convergence is due to their martingale structure. The criterion of non-degeneracy is provided. It differs from those of other random measures constructed as martingale limits of multiplicative processes. It is also very sensitive to small changes in W(t). Interpreting these infinite products in the framework of thermodynamic formalism for random transformations makes these infinite product non-degenerate and convergent via a natural normalization that does not affect non-degenerate original infinite products. The multifractal analysis of the limit measure is studied. It requires suitable Gibbs measures. In the thermodynamic formalism, the notion of weak Gibbs measures was recently introduced and it is associated with a weak principle of bounded variations for the potential function. There, the potential belongs to a subclass of piecewise continuous functions; here, the role of the potential is played by the logarithm of W. A new approach we develop makes it possible to obtain the multifractal nature of infinite random products of harmonics of periodic functions W with a dense countable set of jump points

Multifractal Analysis of inhomogeneous Bernoulli products

We are interested to the multifractal analysis of inhomogeneous Bernoulli products which are also known as coin tossing measures. We give conditions ensuring the validity of the multifractal formalism for such measures. On another hand, we show that these measures can have a dense set of phase transitions

Benoît Mandelbrot Interview 1981

NOTE: to view these items please visit http://dynkincollection.library.cornell.eduInterview conducted by Eugene Dynkin with Benoît B. Mandelbrot in 1981 in in Ithaca, New York

Logique et langage considérés du point de vue de la précorrection des erreurs

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