Showing papers on "Gaussian measure published in 1988"
TL;DR: The best linear unbiased predictors of a random field can be obtained if the covariance function of the random field is specified correctly as discussed by the authors, which is a special case of our problem.
Abstract: Best linear unbiased predictors of a random field can be obtained if the covariance function of the random field is specified correctly. Consider a random field defined on a bounded region $R$. We wish to predict the random field $z(\cdot)$ at a point $x$ in $R$ based on observations $z(x_1), z(x_2), \ldots, z(x_N)$ in $R$, where $\{x_i\}^\infty_{i = 1}$ has $x$ as a limit point but does not contain $x$. Suppose the covariance function is misspecified, but has an equivalent (mutually absolutely continuous) corresponding Gaussian measure to the true covariance function. Then the predictor of $z(x)$ based on $z(x_1), \ldots, z(x_N)$ will be asymptotically efficient as $N$ tends to infinity.
139 citations
TL;DR: In this paper, a transformation for generalized Poisson functionals with the idea of Gaussian white noise was introduced, where the differentiation, renormalization, stochastic integrals, and multiple Wiener integrals were discussed in a way completely parallel with the Gaussian case.
Abstract: Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc ([8], [9]), analogously to the works of T Hida ([3], [4], [5]) Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf [10], [11], [12], [13]) Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc in a way completely parallel with the Gaussian case The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case, as will be stated in Section 5 Conversely, those formulae characterize the types of white noises
98 citations
TL;DR: In this article, the authors studied the quantum dynamics of a particle of mass M in an external potential V(Q), interacting with a simple model environment, a harmonic chain of 2N particles with massm and spring constantk.
Abstract: We study the quantum dynamics of a particle of massM in an external potentialV(Q), interacting with a simple model environment—a harmonic chain of 2N particles with massm and spring constantk. The classical version of this model was studied by Rubin and is equivalent to standard models of a particle interacting with a phonon bath. Settingm=m*/L andk=k*L, we prove that for a suitable class of potentialsV and initial statesω0, the time evolution of the massM particle converges, whenN → ∞ andL → ∞, to the time evolution governed by the Quantum Langevin Equation (QLE) which has been found by Ford, Kac and Mazur. Furthermore we show that, for this class of potentials, the QLE has a unique solution for all positive times, such solution can be expressed as a convergent expansion in the deviation ofV(Q) from a harmonic potential. The equilibrium properties of the particle with massM can be expressed in terms of an integral, over path space, with a Gaussian measure which has mean zero and covariance proportional to\([ - \Delta + \eta h/M\sqrt { - \Delta } ]^{ - 1} \); where\(\eta = 2\sqrt {km} \) is the friction constant, andh is the Plancks' constant (divided by 2π).
30 citations
TL;DR: In this paper, le theoreme de differentiation is shown to be valable dans un espace de Hilbert de dimension infinie avec certaines mesures gaussiennes.
Abstract: On montre que le theoreme de differentiation est valable dans un espace de Hilbert de dimension infinie avec certaines mesures gaussiennes. La demonstration utilise un resultat de l'analyse harmonique concernant le comportement de l'operateur maximal de Hardy-Littlewood dans un espace de dimension elevee
19 citations
01 Jan 1988
14 citations
TL;DR: In this paper, the authors generalized the analysis of Gaussian path-integrals in two and three euclidean dimensions, respectively, by applying the convolution property of quantum-mechanical Green's functions.
10 citations
TL;DR: The Segal-Bargmann space H2(μ) is the space of all entire functions on ℂn that belong to L 2 (μ)-the usual space of Gaussian square-integrable functions as discussed by the authors.
Abstract: Let μ be the Gaussian measure in ℂn given by dμ(z)=(2π)−n exp(−|z|2/2)dV, where dV is the ordinary Lebesgue measure in ℂn. The Segal-Bargmann space H2(μ) is the space of all entire functions on ℂn that belong to L2(μ)-the usual space of Gaussian square-integrable functions. Let P be the orthogonal projection from L2(μ) onto H2(μ). For a measurable function ϕ on ℂn, the multiplication operator Mϕ on L2(μ) is defined by Mϕh =ϕh. The Toeplitz operator Tϕ is defined on H2(μ)by
7 citations
TL;DR: It is proved that nonadaptive information and linear algorithms are optimal and algorithms and information that minimize the average errors lead to the best possible rate of convergence.
Abstract: We study the approximate solution of linear problems in separable Hilbert spaces equipped with a Gaussian measure. We find information and algorithms with the best possible rate of convergence. Although adaptive information and nonlinear algorithms are permitted, we prove that nonadaptive information and linear algorithms are optimal. An algorithm is optimal if it converges with a rate of convergence that is no worse than the rate of any other algorithm except on sets of measure zero. We prove that algorithms and information that minimize the average errors lead to the best possible rate of convergence. This exhibits a close relation between the asymptotic and average case models.
6 citations
TL;DR: In this paper, a uniform estimate of the rate of convergence in the central limit theorem in certain Banach spaces for dependent random variables is established when the Gaussian measure of the ϵ-neighbourhood of the boundary of a set is proportional to ϵ and the third order moment is finite in the strong sense.
5 citations
TL;DR: The issue is analogous to whether sequential designs are more powerful than fixed sample size designs in Bayesian statistics, when observations can be made without noise.
5 citations
TL;DR: An extension of a general theorem on the asymptotic optimality of a linear predictor based on an incorrect covariance function is given and an easily verifiable condition under which the Taylor series predictor given by Bucklew is nearly optimal is obtained.
Abstract: An extension of a general theorem by J.A. Bucklew (ibid., vol. IT-31, 677-679, 1985) on the asymptotic optimality of a linear predictor based on an incorrect covariance function is given. The result is applied to the problem of predicting a small time lag into the future to obtain an easily verifiable condition under which the Taylor series predictor given by Bucklew is nearly optimal. The critical condition of the theorem is as follows: Gaussian measures corresponding to the covariance function used to obtain the predictors and the actual covariance function must be equivalent probability measures (i.e., mutually absolutely continuous measures). >
TL;DR: In this article, the massless singularity of a ferromagnetic Gaussian measure on Ω+ was studied by means of the coarse graining renormalization group method.
Abstract: The massless singularity of a ferromagnetic Gaussian measure on ℤ+ is studied by means of the coarse graining renormalization group method. The result gives information about a singularity behavior of a continued fraction and a time decay rate of a diffusion (random walk) on ℤ+.
TL;DR: In this article, the mean value with respect to a Gaussian measure of the distance between distributions of two finite collections of linear functionals over a Euclidean space with probability measure for certain metrics in the space of finite-dimensional distributions is given.
Abstract: An estimate is given of the mean value with respect to a Gaussian measure of the distance between distributions of two finite collections of linear functionals over a Euclidean space with probability measure for certain metrics in the space of finite-dimensional distributions.
01 Jan 1988
TL;DR: In this paper, the authors consider the Green measure on the infinite dimensional torus T ∞ and show that it is the Wiener semigroup on the Torus T∞ (cf. Heyer,5 Ch. 5).
Abstract: Let {μt}t>0 be the Wiener semigroup on the infinite dimensional torus T∞ (cf. Heyer,5 Ch. 5). Let us consider the Green measure
$$G(dy) = \int\limits_0^\infty {e^{ - t} \mu _t (dy)dt}$$
.
TL;DR: In this article, the authors obtained sufficient conditions for the strong convergence of distributions of functionals of a sequence of stochastic processes, linearly generated by independent random variables, in the case when the distributions of these processes converge weakly to a Gaussian measure.
Abstract: In the present paper we get sufficient conditions for the strong convergence of distributions of functionals of a sequence of stochastic processes, linearly generated by independent random variables, in the case when the distributions of these processes converge weakly to a Gaussian measure.