Gaussian measure

About: Gaussian measure is a research topic. Over the lifetime, 1220 publications have been published within this topic receiving 23177 citations.

Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures

Abstract: We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In particular, we show that such measures have positive isoperimetric constants in the sense of Cheeger and thus always share Poincare-type inequalities. We then describe those log-concave measures which satisfy isoperimetric inequalities of Gaussian type. The results are precised in dimension 1.

On the Characteristic Polynomial of a Random Unitary Matrix

Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞ First we show that \(\), evaluated at a finite set of distinct points, is asymptotically a collection of iid complex normal random variables This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for \(\) For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point There is a phase transition at A= ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy

MCMC methods for diffusion bridges

Abstract: We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parametrised by θ ∈ [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

A probabilistic approach to the geometry of the ℓᵨⁿ-ball

Abstract: This article investigates, by probabilistic methods, various geometric questions on B n p , the unit ball of ln p . We propose realizations in terms of independent random variables of several distributions on B n p , including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B n p . As another application, we compute moments of linear functionals on B n p , which gives sharp constants in Khinchine's inequalities on B n p and determines the 2-constant of all directions on B n p . We also study the extremal values of several Gaussian averages on sections of B n p (including mean width and l-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition.

Approximation, metric entropy and small ball estimates for Gaussian measures

Abstract: A precise link proved by Kuelbs and Li relates the small ball behavior of a Gaussian measure $\mu$ on a Banach space $E$ with the metric entropy behavior of $K_\mu$, the unit ball of the reproducing kernel Hilbert space of $\mu$ in $E$. We remove the main regularity assumption imposed on the unknown function in the link. This enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete Gaussian processes. Moreover, we show that the small ball behavior of a Gaussian process is also tightly connected with the speed of approximation by “finite rank” processes.