ACS Symposium Series

Latała R.

We discuss the method of bounding suprema of canonical processes based on the inclusion of their index set into a convex hull of a well-controlled set of points. While the upper bound is immediate, the reverse estimate was established to date only for a narrow class of regular stochastic processes. We show that for specific index sets, including arbitrary ellipsoids, regularity assumptions may be substantially weakened.

Li J., Tkocz T.

We establish upper and lower bounds with matching leading terms for tails of weighted sums of two-sided exponential random variables. This extends Janson’s recent results for one-sided exponentials.

Mason D.M.

In this note, we prove self-standardized central limit theorems (CLTs) for trimmed subordinated subordinators. We shall see that there are two ways to trim a subordinated subordinator. One way leads to CLTs for the usual trimmed subordinator and a second way to a closely related subordinated trimmed subordinator and CLTs for it.

Bobkov S.G., Roberto C.

We discuss optimal bounds on the Rényi entropies in terms of the Fisher information. In Information Theory, such relations are also known as entropic isoperimetric inequalities.

Oleszkiewicz K.

Motivated by a recent paper of Tanguy (J Theoret Probab 33:692–714, 2020), in which the concept of second-order influences on the discrete cube and Gauss space has been investigated in detail, the present note studies it in a more specific context of Boolean functions on the discrete cube. Some bounds that Tanguy obtained in Tanguy (J Theoret Probab 33:692–714, 2020) as applications of his more general approach are extended and complemented.

Bednorz W.

The first step to study lower bounds for a stochastic process is to prove a special property—Sudakov minoration. The property means that if a certain number of points from the index set are well separated, then we can provide an optimal type lower bound for the mean value of the supremum of the process. Together with the generic chaining argument, the property can be used to fully characterize the mean value of the supremum of the stochastic process. In this article we prove the property for canonical processes based on radial-type log-concave measures.

Koltchinskii V., Wahl M.

Duchemin Q., De Castro Y.

The Random Geometric Graph (RGG) is a random graph model for network data with an underlying spatial representation. Geometry endows RGGs with a rich dependence structure and often leads to desirable properties of real-world networks such as the small-world phenomenon and clustering. Originally introduced to model wireless communication networks, RGGs are now very popular with applications ranging from network user profiling to protein-protein interactions in biology. RGGs are also of purely theoretical interest since the underlying geometry gives rise to challenging mathematical questions. Their resolutions involve results from probability, statistics, combinatorics or information theory, placing RGGs at the intersection of a large span of research communities. This paper surveys the recent developments in RGGs from the lens of high-dimensional settings and nonparametric inference. We also explain how this model differs from classical community-based random graph models, and we review recent works that try to take the best of both worlds. As a by-product, we expose the scope of the mathematical tools used in the proofs.

Merlevède F., Peligrad M.

In this paper we survey some recent progress on the Gaussian approximation for nonstationary dependent structures via martingale methods. First, we present general theorems involving projective conditions for triangular arrays of random variables and then present various applications for rho-mixing and alpha-dependent triangular arrays, stationary sequences in a random time scenery, application to the quenched FCLT, application to linear statistics with alpha-dependent innovations, and application to functions of a triangular stationary Markov chain.

Eisenbaum N., Rosiński J.

The law of a positive infinitely divisible process with no drift is characterized by its Lévy measure on the path space. Based on the recent results of the two authors, it is shown that even for simple examples of such process, the knowledge of their Lévy measures allows to obtain remarkable distributional identities.

Barthe F., Wolff P.

We prove asymptotic estimates for the volume of families of Orlicz balls in high dimensions. As an application, we describe a large family of Orlicz balls which verify a famous conjecture of Kannan, Lovász, and Simonovits about spectral gaps. We also study the asymptotic independence of coordinates on uniform random vectors on Orlicz balls, as well as integrability properties of their linear functionals.

Sambale H.

We prove extensions of classical concentration inequalities for random variables that have α-subexponential tail decay for any α ∈ (0, 2]. This includes Hanson–Wright-type and convex concentration inequalities in various situations. In particular, we show uniform Hanson–Wright inequalities and convex concentration results for simple random tensors in the spirit of recent work by Klochkov–Zhivotovskiy (Electron J Probab 25(22):30, 2020) and Vershynin (Bernoulli 26(4):3139–3162, 2020).

Arras B., Houdré C.

Fradelizi M., Gozlan N., Zugmeyer S.

In this paper, we present a simple proof of a recent result of the second author which establishes that functional inverse Santaló inequalities follow from Entropy-Transport inequalities. Then, using transport arguments together with elementary correlation inequalities, we prove these sharp Entropy-Transport inequalities in dimension 1, which therefore gives an alternative transport proof of the sharp functional Mahler conjecture in dimension 1, for both the symmetric and the general case. We also revisit the proof of the functional inverse Santaló inequalities in the n dimensional unconditional case using these transport ideas.

Grzybowski D.

We prove limit theorems for sums of randomly chosen random variables conditioned on the summands. We consider several versions of the corner-growth setting, including specific cases of dependence among the summands and summands with heavy tails. We also prove a version of Hoeffding’s combinatorial central limit theorem and results for summands taken uniformly from a random sample. These results are proved with concentration of measure techniques.