Vol 238, No 4 (2019)

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).

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.

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.

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 H_T ⊂ L₂ with reproducing kernels K_T (t, s), K_T (t, t) ≤ T.

To any n-dimensional random vector X we may associate its L_p-centroid body Ƶ_p (X) and the corresponding norm. We formulate a conjecture concerning the bound on the Ƶ_p (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.

Let X₁, . . .,X_n be independent nonnegative random variables with cumulative distribution functions F₁, F₂, . . . , F_n satisfying certain (rather mild) conditions. We show that the median of kth smallest order statistic of the vector (X₁, . . . , X_n) is equivalent to the quantile of order (k − 1/2)/n with respect to the averaged distribution \( F=\frac{1}{n}\sum \limits_{i=1}^n{F}_i \).

We study linear images of a symmetric convex body C ⊆ ℝ^N 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 C_N ⊆ ℝ^N, with N → ∞, we obtain new asymptotic results and take first steps to compare with the asymptotic theory.