Hermine Biermé

Bio: Hermine Biermé is an academic researcher from University of Poitiers. The author has contributed to research in topics: Random field & Gaussian. The author has an hindex of 17, co-authored 54 publications receiving 883 citations. Previous affiliations of Hermine Biermé include Paris Descartes University & University of Orléans.

Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants

Abstract: Let fFn : n > 1g be a normalized sequence of random variables in some fixed Wiener chaos associated with a general Gaussian field, and assume that E(F 4

Self-similar random fields and rescaled random balls models

Abstract: We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power-law behavior, we prove that the centered and renormalized random balls field admits a limit with self-similarity properties. Our main result states that all self-similar, translation- and rotation-invariant Gaussian fields can be obtained through a unified zooming procedure starting from a random balls model. This approach has to be understood as a microscopic description of macroscopic properties. Under specific assumptions, we also get a Poisson-type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give L2-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.

Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields

Abstract: Let X = {X(t),t∈ R N } be a Gaussian random field with values in R d defined by X(t )= (X1(t),...,Xd(t)), where X1,...,Xd are independent copies of a centered Gaussian random field X0. Under certain general conditions on X0, we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X �1 (F ), where F ⊆ R d is a nonrandom Borel set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise and the operator-scaling Gaussian random fields with stationary increments constructed �

Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants

Abstract: Let {F_n} be a normalized sequence of random variables in some fixed Wiener chaos associated with a general Gaussian field, and assume that E[F_n^4] --> E[N^4]=3, where N is a standard Gaussian random variable. Our main result is the following general bound: there exist two finite constants c,C>0 such that, for n sufficiently large, c max(|E[F_n^3]|, E[F_n^4]-3) < d(F_n,N) < C max(|E[F_n^3]|, E[F_n^4]-3), where d(F_n,N) = sup |E[h(F_n)] - E[h(N)]|, and h runs over the class of all real functions with a second derivative bounded by 1. This shows that the deterministic sequence max(|E[F_n^3]|, E[F_n^4]-3) completely characterizes the rate of convergence (with respect to smooth distances) in CLTs involving chaotic random variables. These results are used to determine optimal rates of convergence in the Breuer-Major central limit theorem, with specific emphasis on fractional Gaussian noise.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Lévy processes and infinitely divisible distributions

Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Methods Of Modern Mathematical Physics

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