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Processus multistables : Propriétés locales et estimation

Abstract : This PhD thesis deals with some probabilistic, pathwise and statistical properties of multistable stochastic processes, which are tangent at any point to a stable process. Their intensity of jumps and their local regularity are varying with time. We first consider the processes possibly defined as a moving average which are localisable, that is they are tangent to a non-trivial process at any point. We give general conditions which ensure that the moving average is localisable and we characterize the nature of the associated tangent process. We also consider the problem of path synthesis, for which we give both theoretical results and numerical simulations. We present then a different construction of the multistable processes, based on the Ferguson-Klass-LePage series representation. We consider various particular cases of interest, including multistable Levy motion and linear multistable multifractional motion. We study then some probabilistic properties. In particular, we describe the behavior of the incremental moments and the pointwise Hölder exponent. We compute the precise value of the almost sure Hölder exponent in the case of the multistable Levy motion. Finally, we give two estimators of the stability and the localisability functions, and we prove the consistency of those two estimators. We illustrate these convergences with the Levy multistable process and the Linear Multifractional Multistable Motion.
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Contributor : Ronan Le Guével <>
Submitted on : Wednesday, November 3, 2010 - 9:48:13 AM
Last modification on : Monday, March 25, 2019 - 4:52:05 PM
Long-term archiving on: : Friday, February 4, 2011 - 2:48:41 AM


  • HAL Id : tel-00531545, version 1



Ronan Le Guével. Processus multistables : Propriétés locales et estimation. Mathématiques [math]. Université de Nantes, 2010. Français. ⟨tel-00531545⟩



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