Showing papers on "Gaussian measure published in 1990"

On Bayes Procedures

Abstract: In this chapter we describe some of the asymptotic properties of Bayes procedures. These are obtained by using on the parameter set Θ a finite positive measure μ and minimizing the average risk \(\int R(\theta,\rho)\mu(d\theta)\). (See Chapter 2 for notation). The procedure ρ that achieves this minimum will of course depend on the choice of μ. However the literature contains numerous statements to the effect that, for large samples, the choice of μ matters little. This cannot be generally true, but we start with a proposition to this effect. If instead of μ one uses A dominated by μ and if the density \(\frac{d\lambda}{d\mu}\) can be closely estimated, then a procedure that is nearly Bayes for μ is also nearly Bayes for λ.

Grad ø perturbations of massless Gaussian fields

Abstract: We investigate weak perturbations of the continuum massless Gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity.

Bounds on the Efficiency of Linear Predictions Using an Incorrect Covariance Function

Abstract: Suppose $z(\cdot)$ is a random process defined on a bounded set $R \subset \mathbb{R}^1$ with finite second moments. Consider the behavior of linear predictions based on $z(t_1), \ldots, z(t_n)$, where $t_1, t_2, \cdots$ is a dense sequence of points in $R$. Stein showed that if the second-order structure used to generate the predictions is incorrect but compatible with the correct second-order structure, the obtained predictions are uniformly asymptotically optimal as $n \rightarrow \infty$. In the present paper, a general method is described for obtaining rates of convergence when the covariance function is misspecified but compatible with the correct covariance function. When $z(\cdot)$ is Gaussian, these bounds are related to the entropy distance (the symmetrized Kullback divergence) between the measures for the random field under the actual and presumed covariance functions. Explicit bounds are given when $R = \lbrack 0, 1\rbrack$ and $z(\cdot)$ is stationary with spectral density of the form $f(\lambda) = (a^2 + \lambda^2)^{-p}$, where $p$ is a known positive integer and $a$ is the parameter that is misspecified. More precise results are given in the case $p = 1$. An application of this result implies that equally spaced observations are asymptotically optimal in the sense used by Sacks and Ylvisaker in terms of maximizing the Kullback divergence between the actual and presumed models when $z(\cdot)$ is Gaussian.

A note on large deviations for wiener chaos

Abstract: © Springer-Verlag, Berlin Heidelberg New York, 1990, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Representation of Banach Space Valued Martingales as Stochastic Integrals

Abstract: If M = (Mt)t ≥ o is a real-valued, continuous local martingale whose quadratic variation is absolutely continuous relative to Lebesgue measure, then by a theorem of Doob ([7],p.449), M is the stochastic integral of a certain function relative to a Brownian motion-on a possibly extended probability space. This theorem is also well-known in the d-dimensional case (see [12], th.4.5.2). The classical methods of proof can be used without major difficulties to obtain the same theorem for Hilbert space valued, continuous local martingales, but they do not work beyond that case. For continuous local martingales with values in a real separable Banach space, we give a complete different proof of Doob’s theorem. The main application is the representation of Banach space valued Ito processes (defined as in [12]) as solutions of infinite-dimensional stochastic differential equations. This result then can be used to obtain a uniqueness theorem for the so-called martingale problem (in the sense of [12]) in the Banach space case.

The functional of the grand partition function for the investigation of the liquid-gas critical point

Abstract: An approach for the description of a spatially homogeneous system of interacting particles in the vicinity of the critical point is proposed. An explicit expression for the functional integral form of the partition function in the grand canonical ensemble is obtained. The functional integral is defined on a set of the collective variables (CV) {ϱk}. The variable ϱk corresponds to the density vibration mode with the wave vector k. It is shown that in the vicinity of the critical point, for all k larger than some B, the CV are distributed with the Gaussian measure density. In the long wave region, k < B, the surfaces of the cumulants consisting of the correlation functions of some reference system (the reference system is a model system of particles with repulsive pair interactions) possess wide plateaus in the vicinity of the point k = 0. It is shown that due to these two facts the homogeneous system can be put in correspondance with a certain lattice system with spacing c = πB. The problem of the calculation of the partition function can be reduced to the calculation of the functional for a three-dimensional Ising model with an external field.

Gaussian Measure of Translated Balls in a Banach Space

Abstract: On etudie la differentiabilite au sens de Gâteaux d'une mesure gaussienne de boules translatees dans un espace de Banach

Some Probability and Entropy Estimates for Gaussian Measures

Abstract: We compare two estimates for the measure of Banach neighborhoods of Hilbert balls in the reproducing kernel space. Borell’s estimate [1] is quite general and is known to be sharp for certain cases which involve small probabilities. However, Talagrand [7] and Goodman [4] use the openness of certain sets to obtain alternative estimates for cases in which the probability is near one.

Bosonic Strings and Measures on Infinite Dimensional Manifolds

Abstract: This paper reports and extends the results of [1] in which a correspondence was set up between the Polyakov model for dimension of space — time d ≤ 12 and the Liouville model studied in [25]. With the aim of further improving the mathematical framework underlying the reduction of the Polyakov measure to the Liouville measure already set up in [1], it gives a precise description of the formal steps of the reduction procedure in the context of Gaussian measures on infinite dimensional manifolds.

Gaussian quantization of the Klein-Gordon equation in variational derivatives and secondary quantization of a Boson string

Abstract: A Klein-Gordon equation in variational derivatives is proposed for the quantum theory of a Boson string in a fixed gauge. By using this equation a secondary quantization is constructed for a Boson string in the functional space of string functionals that are square summable in a countable additive Gaussian measure.

Analog of heat equation for gaussian measure of a ball in Hilberrt space

Abstract: If μ a,T is a Gaussian measure on a Hilbert space with meana and covariance operatorT, andr is a fixed positive number, then the functiong(a,T)=μ a,T {‖x‖