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- W2397030198 abstract "In this thesis, local regularity properties of some multiparameter, set-indexed and eventually L2-indexed random fields are investigated. The goal is to extend standard tools of the theory of stochastic processes, in particular local Holder regularity, to indexing collection which are not totally ordered.The classic Kolmogorov continuity criterion gives a lower estimate of the Holder regularityof a stochastic process indexed by a subset of R or RN . Using the lattice structure of the indexing collections in the theory of set-indexed processes of Ivanoff and Merzbach, Kolmogorov’scriterion is extended to this framework. Different increments for set-indexed processes are considered,and several Holder exponents are defined accordingly. For Gaussian processes, these exponents are, almost surely and uniformly along the sample paths, deterministic and related to the law of the increments of the process. This is applied to the set-indexed fractional Brownian motion, for which the regularity is constant. In order to exhibit a process having a variable regularity,we resorted to structures of Abstract Wiener Spaces, and defined a fractional Brownian field indexed by a product space (0, 1=2]_L2(T,m), based on a family of positive definite kernels kh, h 2 (0, 1=2]. This field encompasses a large class of existing multiparameter fractional Brownian processes, which are exhibited by choosing appropriate metric spaces (T,m). It is proven that the law of the increments of such a field is bounded above by a function of the increments in both parameters of the field. Applying the techniques developed to measure the local Holder regularity, it is proven that this field can lead to a set-indexed, or L2-indexed, Gaussian process with prescribed local regularity.The last part is devoted to the study of the singularities induced by the multiparameter process defined by the covariance kh on L2([0, 1]_,dx). This process is a natural extension of the fractional Brownian motion and of the Brownian sheet. At the origin 0 of RN+, this multiparameter fractional Brownian motion has a different regularity behaviour. A Chung (or lim inf ) law of the iterated logarithm permits to observe this." @default.
- W2397030198 created "2016-06-24" @default.
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- W2397030198 date "2014-09-29" @default.
- W2397030198 modified "2023-10-16" @default.
- W2397030198 title "Local regularity of some fractional Brownian fields" @default.
- W2397030198 hasPublicationYear "2014" @default.
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