Some convergence results on quadratic forms for random fields and application to empirical covariances

Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. We obtain a non central limit theorem under a minimal integrability condition, which allows isotropic and anisotropic models. We apply our limit theorems and those of Ginovian (J. Contemp. Math. Anal. 34(2):1?C15) to obtain the asymptotic behavior of the empirical covariances of Gaussian fields, which is a particular example of quadratic forms. We show that it is possible to obtain a Gaussian limit when the spectral density is not in L ². Therefore the dichotomy observed in dimension d?=?1 between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in d?>?1.     

The norm of polynomials in large random and deterministic matrices

Let ${{\bf X}_N =(X_1^{(N)}, \ldots, X_p^{(N)})}$ be a family of N × N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices ${{\bf Y}_N =(Y_1^{(N)}, \ldots, Y_q^{(N)})}$ , possibly random but independent of X _N , for which the operator norm of ${P({\bf X}_N, {\bf Y}_N, {\bf Y}_N^*)}$ converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y _N and of the polynomials P, we get for a large class of matrices the ??no eigenvalues outside a neighborhood of the limiting spectrum?? phenomena. We give examples of diagonal matrices Y _N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

Résolution numérique d'un problème elliptique fortement anisotrope en deux dimensions par une méthode de paramétrisation

We introduce a numerical method for solving an anisotropic elliptic problem. We address the case where the direction of the anisotropy varies, and the anisotropy is high. A finite volume scheme is implemented to solve the problem for small anisotropy ratio, then the parameterization method consists in devising an extrapolation of the solution of the anisotropic problem by combining solutions of a sequence of isotropic problems. To cite this article: P. Guillaume, V. Latocha, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Normes des extensions quaternionique d'opérateurs réels

We consider bounded linear operators defined on real normed spaces, and with range in quaternionic spaces. We study the norms of the quaternionic extensions of such operators. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Estimateur crible de l'opérateur d'un processus ARB(1)

We consider the sieve estimator of the operator of a Banach autoregressive process. We show the almost sure convergence when the operator is 2-summing, strictly 2-integral, afterwards 2-nuclear for the adequate norms. To cite this article: F. Rachedi, T. Mourid, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Error-free computer solution of certain systems of linear equations

In this article we propose a procedure which generates the exact solution for the system Ax = b, where A is an integral nonsingular matrix and b is an integral vector, by improving the initial floating-point approximation to the solution. This procedure, based on an easily programmed method proposed by Aberth [1], first computes the approximate floating-point solution x^* by using an available linear equation solving algorithm. Then it extracts the exact solution x from x^* if the error in the approximation x^* is sufficiently small. An a posteriori upper bound for the error of x^* is derived when Gaussian Elimination with partial pivoting is used. Also, a computable upper bound for |det(A)|, which is an alternative to using Hadamard's inequality, is obtained as a byproduct of the Gaussian Elimination process.     

Hypersurfaces d'un fibré vectoriel Riemannien à courbure de Gauss prescriteHypersurfaces of a Riemannian vector bundle with prescribed Gaussian curvature

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and Σ the sphere subbundle of E. We look for embeddings of Σ into E admitting prescribed Gaussian curvatures of various types. To cite this article: A. Hanani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 927–930.     

Invariant scale matrix hypothesis tests under elliptical symmetry

The usual assumption in multivariate hypothesis testing is that the sample consists of n independent, identically distributed Gaussian m-vectors. In this paper this assumption is weakened by considering a class of distributions for which the vector observations are not necessarily either Gaussian or independent. This class contains the elliptically symmetric laws with densities of the form f(X(n × m)) = ψ[tr(X ? M)′ (X ? M)Σ^?1]. For testing the equality of k scale matrices and for the sphericity hypothesis it is shown, by using the structure of the underlying distribution rather than any specific form of the density, that the usual invariant normal-theory tests are exactly robust, for both the null and non-null cases, under this wider class.     

Un théorème limite pour les covariances des spins en deux sites dans le modèle de Sherrington–Kirkpatrick avec champ externe

We give some properties of spin covariances in the case of the Sherrington–Kirkpatrick model with an external field; a non Gaussian limit theorem for those covariances is proved. To cite this article: A. Hanen, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Operator norms,multiplicativity factors,and C-numerical radii

Let V be a normed vector space over C, let $\text{B}$(V) denote the algebra of linear bounded operators on V, and let N be an arbitrary seminorm or norm on $\text{B}$(V). In this paper we discuss multiplicativity factors for N, i.e., constants μ>0 for which N_μ≡μN is submultiplicative. We find that, while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors, in the infinite dimensional case N may or may not have such factors. Our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius, called C-numerical radii. This is done with the help of a combinatorial inequality which seems to be of independent interest.     

Characterization of fuzzy measures constructed by means of triangular norms

Following the ideas presented by the author (E. P. Klement, J. Math. Anal. Appl.85 (1982), 543–565) finite T-fuzzy measures are introduced, T being a measurable triangular norm. We show that a T-fuzzy measure is always a fuzzy measure, as considered earlier (E. P. Klement, J. Math. Anal. Appl.25 (1980), 330–339). Then we study the relation to the integral with respect to some classical measure. Finally, for some special triangular norms T, we give precise characterizations of the corresponding classes of T-fuzzy measures.     

Optimal Gaussian Sobolev embeddings

A reduction theorem is established, showing that any Sobolev inequality, involving arbitrary rearrangement-invariant norms with respect to the Gauss measure in Rn, is equivalent to a one-dimensional inequality, for a suitable Hardy-type operator, involving the same norms with respect to the standard Lebesgue measure on the unit interval. This result is exploited to provide a general characterization of optimal range and domain norms in Gaussian Sobolev inequalities. Applications to special instances yield optimal Gaussian Sobolev inequalities in Orlicz and Lorentz(-Zygmund) spaces, point out new phenomena, such as the existence of self-optimal spaces, and provide further insight into classical results.     

Stochastic integration with respect to Gaussian processesIntégrale stochastique pour les processus gaussiens

We construct a Stratonovitch–Skorohod-like stochastic integral for general Gaussian processes. We study its sample path regularity and one of its numerical approximating schemes. We also analyze the way it is transformed by an absolutely continuous change of probability and we give an Itô formula. To cite this article: L. Decreusefond, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 903–908.     

Heat kernels for non-divergence operators of Hörmander type

We prove the existence of a fundamental solution for a class of Hörmander heat-type operators. For this fundamental solution and its derivatives we obtain sharp Gaussian bounds that allow to prove an invariant Harnack inequality. To cite this article: M. Bramanti et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Un schéma linéaire vérifiant le principe du maximum pour des opérateurs de diffusion très anisotropes sur des maillages déformés

We introduce a generalized finite-difference method for anisotropic diffusion operators on distorted grids. We calculate the second-order derivatives in space using a Taylor expansion. The resulting global matrix associated to the scheme is an M-matrix. Thanks to a certain assumption on the grid properties, we show the convergence of the scheme. We show the robustness of the method in comparison with analytical solutions and results obtained by other numerical schemes. To cite this article: C. Le Potier, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A central limit theorem for D(A)-valued processes

Let D(A) be the space of set-indexed functions that are outer continuous with inner limits, a generalization of D[0, 1]. This paper proves a central limit theorem for triangular arrays of independent D(A) valued random variables. The limit processes are not restricted to be Gaussian, but can be quite general infinitely divisible processes. Applications of the theorem include construction of set-indexed Lévy processes and a unified central limit theorem for partial sum processes and generalized empirical processes. Results obtained are new even for the D[0, 1] case.     

Extreme values of particular non-linear processesValeurs extrêmes pour des processus non linéaires particuliers

We investigate the asymptotic behavior of the maxima of a general class of deterministic chaotic processes – including the tent map and the logistic map –, of noisy chaotic processes, and of the Gaussian long memory k-factor Gegenbauer processes. To cite this article: D. Guégan, S. Ladoucette, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 73–78.     

Une généralisation de l'intégrale stochastique de Wick–Itô

The covariance of the fractional Brownian motion belongs to a family of positive functions introduced by Schoenberg in the 1930s. We show that one can define a stochastic integral for a large sub-family of the corresponding Gaussian second order stochastic processes. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

High extrema of Gaussian chaos processes

Let ξ ( t)=(ξ ₁(t),…,ξ _d (t)) be a Gaussian stationary vector process. Let \(g:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}\) be a homogeneous function. We study probabilities of large extrema of the Gaussian chaos process g(ξ(t)). Important examples include \(g(\mathbf {\boldsymbol {\xi }}(t))={\prod }_{i=1}^{d}\xi _{i}(t)\) and \(g(\mathbf {\boldsymbol {\xi }}(t))={\sum }_{i=1}^{d}a_{i}{\xi _{i}^{2}}(t)\). We review existing results partially obtained in collaboration with E. Hashorva, D. Korshunov, and A. Zhdanov. We also present the principal methods of our investigations which are the Laplace asymptotic method and other asymptotic methods for probabilities of high excursions of Gaussian vector process’ trajectories.