A diffusion approximation result for two parameter processes

Summary We consider a one-dimensional linear wave equation with a small mean zero dissipative field and with the boundary condition imposed by the so-called Goursat problem. In order to observe the effect of the randomness on the solution we perform a space-time rescaling and we rewrite the problem in a diffusion approximation form for two parameter processes. We prove that the solution converges in distribution toward the solution of a two-parameter stochastic differential equation which we identify. The diffusion approximation results for oneparameter processes are well known and well understood. In fact, the solution of the one-parameter analog of the problem we consider here is immediate. Unfortunately, the situation is much more complicated for two-parameter processes and we believe that our result is the first one of its kind.Partially supported by ONR N00014-91-J-1010     

Remarques sur les m\'ethodes de projection pour l'approximation des \'equations de Navier--Stokes

Summary. L'objet de cet article est de montrer que les estimations de convergence sur la pression pour les m\'ethodes de projection d\'ecrites dans \cite{Shen1} et \cite{Shen2} ne sont pas obtenues correctement car elles sont toutes bas\'ees sur une in\'egalit\'e fausse. Il semble qu'on ait besoin d'une convergence en de la vitesse dans pour d\'emontrer les estimations de convergence sur la pression en . La question de savoir si la m\'ethode de projection a un taux de convergence pour la pression plus \'elev\'e que reste ouverte. Received June 1, 1993     

Coupling for Markov renewal processes and the rate of convergence in ergodic theorems for processes with semi-Markov switchings

Coupling procedures for Markov renewal processes are described. Applications to ergodic theorems for processes with semi-Markov switchings are considered.This paper was partly prepared with the support of NFR Grant F-UP 10257-300.     

The drift of a one-dimensional self-repellent random walk with bounded increments

Summary Consider a one-dimensional walk (S _k ) _k having steps of bounded size, and weight the probability of the path with some factor 1–(0,1) for every single self-intersection up to timen. We prove thatS _n /S S converges towards some deterministic number called the effective drift of the self-repellent walk. Furthermore, this drift is shown to tend to the basic drift as tends to 0 and, as tends to 1, to the self-avoiding walk's drift which is introduced in [10]. The main tool of the present paper is a representation of the sequence of the local times as a functional of a certain Markov process.Partially supported by Swiss National Sciences Foundation Grant 20-36305.92     

On Toeplitz type quadratic functionals of stationary Gaussian processes

Summary A central limit theorem for Toeplitz type quadratic functionals of a stationary Gaussian processX(t),t, is proved, generalizing the result of Avram [1] for discrete time processes. The result is applied to the problem of nonparametric estimation of linear functionals of an unknown spectral density function. We give some upper bounds for the minimax mean square risk of the nonparametric estimators, similar to those by Ibragimov and Has'minskii [12] for a probability density function.     

The non-existence of symplectic multi-derivative Runge-Kutta methods

A sufficient condition for the symplecticness ofq-derivative Runge-Kutta methods has been derived by F. M. Lasagni. In the present note we prove that this condition can only be satisfied for methods withq1, i.e., for standard Runge-Kutta methods. We further show that the conditions of Lasagni are also necessary for symplecticness so that no symplectic multi-derivative Runge-Kutta method can exist.This research has been supported by project PB89-0351 (Dirección General de Investigación Científica y Técnica) and by project No. 20-32354.91 of Swiss National Science Foundation.     

Local convergence of the (exact and inexact) iterative aggregation method for linear systems and Markov operators

Summary. The iterative aggregation method for the solution of linear systems is extended in several directions: to operators on Banach spaces; to the method with inexact correction, i.e., to methods where the (inner) linear system is in turn solved iteratively; and to the problem of finding stationary distributions of Markov operators. Local convergence is shown in all cases. Convergence results apply to the particular case of stochastic matrices. Moreover, an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled Markov chains, as well as for Markov chains with other zero-nonzero structures. Received May 25, 1991/Revised version received February 23, 1994     

On the Durrmeyer-type modification of some discrete approximation operators

In [10], for continuous functionsf from the domain of certain discrete operatorsL _n the inequalities are proved concerning the modulus of continuity ofL _nf. Here we present analogues of the results obtained for the Durrmeyer-type modification $\tilde L_n $ ofL _n. Moreover, we give the estimates of the rate of convergence of $\tilde L_n f$ in Hölder-type norms     

Backward errors for eigenvalue and singular value decompositions

Summary. We present bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two-norm and in the Frobenius norm. Through different orthogonal decompositions of the computed eigenvectors we can define different symmetric backward errors for the eigenvalue decomposition. When the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small. Consequently it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured. Analogous results hold for the singular vectors. We indicate the effect of our error bounds on implementations for eigenvector and singular vector computation. In a more general context we prove that the distance of an appropriately scaled matrix to its orthogonal QR factor is not much larger than its distance to the closest orthogonal matrix. Received July 19, 1993     

Large deviation principle for stochastic evolution equations

Summary The large deviation principle obtained by Freidlin and Wentzell for measures associated with finite-dimensional diffusions is extended to measures given by stochastic evolution equations with non-additive random perturbations. The proof of the main result is adopted from the Priouret paper concerning finite-dimensional diffusions. Exponential tail estimates for infinite-dimensional stochastic convolutions are used as main tools.