Topic
Gaussian measure
About: Gaussian measure is a research topic. Over the lifetime, 1220 publications have been published within this topic receiving 23177 citations.
Papers published on a yearly basis
Papers
More filters
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 new definition of the Gaussian multiplicative chaos and an approach based on the relation of subcritical Gaussians to randomized shifts of a Gaussian measure were proposed.
137 citations
TL;DR: In this article, the authors studied the Toeplitz operators on the Segal-Bargmann spaces of Gaussian measure square-integrable entire functions on complex n-space C n.
134 citations
TL;DR: In this paper, the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions was studied, and the expected volume of the nodal set was shown to be bounded by O(1/δ) by Gaussian probability measure on the eigenspaces.
Abstract: We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4π2E with growing multiplicity \({\mathcal{N}} \rightarrow \infty\), and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is \(const{\sqrt{E}}\). Our main result is that the variance of the volume normalized by \(\sqrt{E}\) is bounded by \(O(1/\sqrt{{\mathcal{N}}})\), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.
132 citations
TL;DR: In this article, a polynomial chaos expansion is used to represent a Gaussian process in terms of multidimensional polynomials orthogonal with respect to the Gaussian measure.
Abstract: A procedure is presented in this paper for developing a representation of lognormal stochastic processes via the polynomial chaos expansion. These are processes obtained by applying the exponential operator to a gaussian process. The polynomial chaos expansion results in a representation of a stochastic process in terms of multidimensional polynomials orthogonal with respect to the gaussian measure with the dimension defined through a set of independent normalized gaussian random variables. Such a representation is useful in the context of the spectral stochastic finite element method, as well as for the analytical investigation of the mathematical properties of lognormal processes.
130 citations