卷 238, 编号 4 (2019)
- 年: 2019
- 文章: 16
- URL: https://journals.rcsi.science/1072-3374/issue/view/14999
Article
From Editorial Board
A Few Recollections
Large Deviations for Level Sets of a Branching Brownian Motion and Gaussian Free Fields
摘要
We study deviation probabilities for the number of high positioned particles in branching Brownian motion and confirm a conjecture of Derrida and Shi. We also solve the corresponding problem for the two-dimensional discrete Gaussian free field. Our method relies on an elementary inequality for inhomogeneous Galton–Watson processes.
Gaussian Mixtures and Normal Approximation for V. N. Sudakov’s Typical Distributions
摘要
We derive a general upper bound on the distance of the standard normal law to typical distributions in V. N. Sudakov’s theorem (in terms of the weighted total variation).
On the Equality of Values in the Monge and Kantorovich Problems
摘要
This paper is concerned with the study of conditions under which the Monge and Kantorovich problems with a continuous cost function on a product of two completely regular spaces and two given atomless Radon measures-projections on these spaces have equal values of the corresponding infima.
Duality and Free Measures in Vector Spaces, the Spectral Theory of Actions of Non-Locally Compact Groups
摘要
The paper presents a general duality theory for vector measure spaces taking its origin in author’s papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications.
On an Exponential Functional for Gaussian Processes and Its Geometric Foundations
摘要
After setting geometric notions, we revisit an exponential functional which has arisen in several contexts, with special attention to a set of geometric parameters and associated inequalities.
An Optimal Transport Approach for the Kinetic Bohmian Equation
摘要
We study the existence theory of solutions of the kinetic Bohmian equation, a nonlinear Vlasov-type equation proposed for the phase-space formulation of Bohmian mechanics. Our main idea is to interpret the kinetic Bohmian equation as a Hamiltonian system defined on an appropriate Poisson manifold built on a Wasserstein space. We start by presenting an existence theory for stationary solutions of the kinetic Bohmian equation. Afterwards, we develop an approximative version of our Hamiltonian system in order to study its associated flow. We then prove the existence of solutions of our approximative version. Finally, we present some convergence results for the approximative system; our aim is to show that, in the limit, the approximative solution satisfies the kinetic Bohmian equation in a weak sense.
Deviation Inequalities for Convex Functions Motivated by the Talagrand Conjecture
摘要
Motivated by Talagrand’s conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and, in particular, by its continuous analogue involving regularization properties of the Ornstein–Uhlenbeck semigroup acting on integrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.
On Estimation of Functions of a Parameter Observed in Gaussian Noise
摘要
The main problem of the paper looks as follows. A functional parameter θ ∈ Θ ⊂ L2(−∞,∞) is observed in Gaussian noise. The problem is to estimate the value F(θ) of a given function F. A construction of asymptotically efficient estimates for F(θ) is suggested under the condition that Θ admits approximations by subspaces HT ⊂ L2 with reproducing kernels KT (t, s), KT (t, t) ≤ T.
Gaussian Approximation Numbers and Metric Entropy
摘要
The aim of this paper is to survey properties of Gaussian approximation numbers. We state the basic relations between these numbers and other s-numbers as, e.g., entropy, approximation, or Kolmogorov numbers. Furthermore, we fill a gap and prove new two-sided estimates in the case of operators with values in a K-convex Banach space. In the final section, we apply relations between Gaussian and other s-numbers to the d-dimensional integration operator defined on L2[0, 1]d.
On Ƶp-Norms of Random Vectors
摘要
To any n-dimensional random vector X we may associate its Lp-centroid body Ƶp (X) and the corresponding norm. We formulate a conjecture concerning the bound on the Ƶp (X)-norm of X and show that it holds under some additional symmetry assumptions. We also relate our conjecture to estimates of covering numbers and Sudakov-type minoration bounds.
On Optimal Matching of Gaussian Samples
摘要
Let X1, . . .,Xn be independent random variables having as common distribution the standard Gaussian measure μ on ℝ2 and let \( {\mu}_n=\frac{1}{n}\sum \limits_{i=1}^n{\delta}_{X_i} \) be the associated empirical measure. We show that
\( \frac{1}{C}\frac{\log n}{n}\le \) ???? \( \left({\mathrm{W}}_2^2\left({\mu}_n,\mu \right)\right)\le C\frac{{\left(\log n\right)}^2}{n} \)
for some numerical constant C > 0, where W2 is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra, and D. Trevisan.
Estimates for Order Statistics in Terms of Quantiles
摘要
Let X1, . . .,Xn be independent nonnegative random variables with cumulative distribution functions F1, F2, . . . , Fn satisfying certain (rather mild) conditions. We show that the median of kth smallest order statistic of the vector (X1, . . . , Xn) is equivalent to the quantile of order (k − 1/2)/n with respect to the averaged distribution \( F=\frac{1}{n}\sum \limits_{i=1}^n{F}_i \).
A Sharp Rate of Convergence for the Empirical Spectral Measure of a Random Unitary Matrix
摘要
We consider the convergence of the empirical spectral measures of random N × N unitary matrices. We give upper and lower bounds showing that the Kolmogorov distance between the spectral measure and uniform measure on the unit circle is of order log N/N, both in expectation and almost surely. This implies, in particular, that the convergence happens more slowly for Kolmogorov distance than for the L1-Kantorovich distance. The proof relies on the determinantal structure of the eigenvalue process.
Gaussian Convex Bodies: a Nonasymptotic Approach
摘要
We study linear images of a symmetric convex body C ⊆ ℝN under an n × N Gaussian random matrix G, where N ≥ n. Special cases include common models of Gaussian random polytopes and zonotopes. We focus on the intrinsic volumes of GC and study the expectation, variance, small and large deviations from the mean, small ball probabilities, and higher moments. We discuss how the geometry of C, quantified through several different global parameters, affects such concentration properties. When n = 1, G is simply a 1 × N row vector, and our analysis reduces to Gaussian concentration for norms. For matrices of higher rank and for natural families of convex bodies CN ⊆ ℝN, with N → ∞, we obtain new asymptotic results and take first steps to compare with the asymptotic theory.