arXiv subjects
Publications and source records attributed to Sergey G. Bobkov.
We discuss variants of construction of measurable subgradients for multivariate convex functions and the problem of characterization of the $Δ_2$-condition in terms of their directional derivatives. Furthermore we study related basic properties of Luxemburg and Orlicz pseudo-norms for vector-valued functions.
An upper bound for the Kantorovich transport distance between probability measures on multidimensional Euclidean spaces is given in terms of transport distances between one dimensional projections. This quantifies the Cramér-Wold continuity theorem for the weak convergence of probability measures.
Pinsker-type inequalities are considered for the weighted total variation distance between probability measures in terms of the Rényi divergence powers. They are applied in derivation of transport-entropy inequalities under moment-type conditions.
An upper bound for Zolotarev distances between probability measures on multidimensional Euclidean spaces is given in terms of similar distances between one dimensional projections.
Basic general properties are considered for the Fisher-type information involving higher order derivatives. They are used to explore various properties of probability densities and to derive Stam-type inequalities.
For normalized sums $Z_n$ of i.i.d. random variables, we explore necessary and sufficient conditions which guarantee the normal approximation with respect to the Rényi divergence of infinite order. In terms of densities $p_n$ of $Z_n$, this is a strengthened variant of the local limit theorem taking the form $\sup_x (p_n(x) - φ(x))/φ(x) \rightarrow 0$ as $n \rightarrow \infty$.
Covariance representations are developed for the uniform distributions on the Euclidean spheres in terms of spherical gradients and Hessians. They are applied to derive a number of Sobolev type inequalities and to recover and refine the concentration of measure phenomenon, including second order concentration inequalities. A detail account is also given in the case of the circle, with a short overview of Höffding's kernels and covariance identities in the class of periodic functions.
We start with a brief survey on Höffding's kernels, its properties, related spectral decompositions, and discuss marginal distributions of Höffding measures. In the second part of this note, one-dimensional covariance representations are considered over compactly supported probability distributions in the class of periodic smooth functions. Höffding's kernels are used in the construction of mixing measures whose marginals are multiples of given probability distributions, leading to optimal kernels in periodic covariance representations.
We revisit several results on exponential integrability in probability spaces and derive some new ones. In particular, we give a quantitative form of recent results by Cianchi-Musil and Pick in the framework of Moser-Trudinger-type inequalities, and recover Ivanisvili-Russell's inequality for the Gaussian measure. One key ingredient is the use of a dual argument, which is new in this context, that we also implement in the discrete setting of the Poisson measure on integers.
We discuss a natural extension of Gilles Pisier's approach to the study of measure concentration, isoperimetry and Poincaré-type inequalities. This approach allows one to explore counterparts of various results about Gaussian measure in the class of rotationally invariant probability distribution on Euclidean spaces, including multidimensional Cauchy measures.
Two-sided bounds are explored for concentration functions and Rényi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.
Convergence of order $O(1/\sqrt{n})$ is obtained for the distance in total variation between the Poisson distribution and the distribution of the number of fixed size cycles in generalized random graphs with random vertex weights. The weights are assumed to be independent identically distributed random variables which have a power-law distribution. The proof is based on the Chen--Stein approach and on the derived properties of the ratio of the sum of squares of random variables and the sum of these variables. These properties can be applied to other asymptotic problems related to generalized random graphs.
Two--sided bounds are constructed for a probability density function of a weighted sum of chi-square variables. Both cases of central and non-central chi-square variables are considered. The upper and lower bounds have the same dependence on the parameters of the sum and differ only in absolute constants. The estimates obtained will be useful, in particular, when comparing two Gaussian random elements in a Hilbert space and in multidimensional central limit theorems, including the infinite-dimensional case.
We explore an asymptotic behavior of entropies for sums of independent random variables that are convolved with a small continuous noise.
We explore an asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.
We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order $d-1$ for any $d \in \mathbb{N}$. The bounds are based on $d$-th order derivatives or difference operators. In particular, we consider deviations of functions of independent random variables and differentiable functions over probability measures satisfying a logarithmic Sobolev inequality, and functions on the unit sphere. Applications include concentration inequalities for $U$-statistics as well as for classes of symmetric functions via polynomial approximations on the sphere (Edgeworth-type expansions).
We explore an asymptotic behavior of Rényi entropy along convolutions in the central limit theorem with respect to the increasing number of i.i.d. summands. In particular, the problem of monotonicity is addressed under suitable moment hypotheses.
Let $F_n$ denote the distribution function of the normalized sum $Z_n = (X_1 + \dots + X_n)/σ\sqrt{n}$ of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $F_n$ to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of $F_n$ by the Edgeworth corrections (modulo logarithmically growing factors in $n$) are given in terms of the characteristic function of $X_1$. Particular cases of the problem are discussed in connection with Diophantine approximations.