arXiv subjects
Publications and source records attributed to Witold Bednorz.
We prove a prescribed-index-law comparison for canonical processes with independent symmetric coordinates having log-concave tails, with no doubling assumption
We extend the recent result of Park and Pham concerning the positive selector process to canonical processes generated by i.i.d. nonnegative random variables satisfying minimal tail assumptions. We also provide a result of the same nature for canonical processes based on general i.i.d. positive variables.
We provide an alternative proof of the recent result by Park and Pham (2022) on the expected suprema of positive selector and empirical processes. We extend it to positive infinitely divisible processes.
We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a càdlàg path and an occupation measure associated with this path. Using this result we define local times of fractional Brownian motions (classically defined as densities of relevant occupation measure) as weak limits of properly normalized numbers of interval crossings. We also discuss a similar result for càdlàg semimartingales, in particular for alpha-stable processes, and for Rosenblatt processes, and provide natural examples of deterministic paths which possess quadratic or higher order variation but no local times.
In this paper we improve the best known constant for the discrepancy formulated in the Komlos Conjecture. The result is based on the improvement of the subgaussian bound for the random vector constructed in the Gram-Schmidt Random Walk algorithm. Moreover, we present detailed argument for the smoothed analysis of this random vector. The analysis concerns a modification of a given matrix in the conjecture by a Gaussian type perturbation. Our result improves the recent paper in this direction.
In this paper we complete the full characterization of the expected suprema of infinitely divisible processes. In particular, we remove the technical assumption called $H(C_{0},δ)$ condition and settle positively the conjecture posed by M. Talagrand.
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.
In this paper we consider the existence of weakly càdlàg versions of a solution to a linear equation in a Hilbert space $H$, driven by a Levy process taking values in a Hilbert space $U$. In particular we are interested in diagonal type processes, where process on coordinates are functionals of independent $α$ stable symmetric process. We give the if and only if characterization in this case. We apply the same techniques to obtain a sufficient condition for existence of a càdlàg versions of stable processes described as integrals of deterministic functions with respect to symmetric $α$-stable random measures with $α\in[1,2)$.
In this paper we prove a Lévy-Ottaviani type of property for the Bernoulli process defined on an interval. Namely, we show that under certain conditions on functions $(a_i)_{i=1}^{n}$ and for independent Bernoulli random variables $(\varepsilon_i)_{i=1}^{n}$, $\mathbb{P}(\sup_{t\in [0,1]}\sum^n_{i=1}a_i(t)\varepsilon_i\geq c)$ is dominated by $C\mathbb{P}(\sum^n_{i=1}a_i(1)\varepsilon_i\geq1)$, where $c$ and $C$ are explicit numerical constants independent of $n$. The result is a partial answer to the conjecture of W. Szatzschneider that the domination holds with $c=1$ and $C=2$.
In this paper we improve Bernoulli comparison. The result works for independent Rademacher random variables $(\varepsilon_i)_{i\geq1}$ and states that we can compare $\mathbb{E}\sup_{t\in T}\sum_{i\geq1}φ_{i}(t)\varepsilon_i$ with $\mathbb{E}\sup_{t\in T}\sum_{i\geq1}t_i\varepsilon_i$, where a function $φ=(φ_i)_{i\geq1}: \ell^2\supset T\rightarrow\ell^2$, satisfies certain conditions. Originally, it is assumed that each of $φ_i$ is a contraction. We relax this assumption towards comparison of Gaussian parts of increments, which can be described in the following way. For all $s,t\in T$, $p\geq 0$ $$ \inf_{|I^c|\leq Cp}\sum_{i\in I}|φ_i(t)-φ_i(s)|^2\leq C^2\inf_{|I^c|\leq p}\sum_{i\in I}|t_i-s_i|^2, $$ where $C\geq 1$ is an absolute constant and $I\subset\mathbb{N}$, $I^c=\mathbb{N}\backslash I$.
Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a family of ruled surfaces. We show their usefulness in visualizing the process using commonly available modeling software.
Kwapien and Woyczynski asked in their monograph (1992) whether their notion of superstrong domination is inherited when taking sums of independent symmetric random vectors (one vector dominates another if, essentially, tail probabilities of any norm of the two vectors compare up to some scaling constants). We answer this question positively. As a by-product of our methods, we establish that a certain notion of weak concentration is also preserved by taking sums of independent symmetric random vectors.
In this paper we study the question how to easily verify that the expectation of the supremum of a one canonical Bernoulli process dominates the same quantity for another process of this type. In the setting of Gaussian canonical processes it is known that such a comparison holds for contractions in the Euclidean distance. We do state and prove a similar result for Bernoulli processes. In particular we get a partial answer to the Oleszkiewicz conjecture about the comparability of weak and strong moments for type Bernoulli series in a Banach space.
In this paper we show an alternative approach to the concentration of truncated variation for stochastic processes on a real line. Our method is based on the moments control and can be used to generalize the results to the case of processes with heavy tails.
In this paper we discuss the question how to bound supremum of a stochastic process with the index set of a product type. There is a tempting idea to approach the question by the analysis of the process on each of the marginal index spaces separately. However it turns out that we also need to study suitable partitions of the whole index space. We show what can be done in this direction and how to use the method to reprove some known results. In particular we observe that all known applications of the Bernoulli Theorem can be obtained in this way, moreover we use the shattering dimension to slightly extend the application to VC classes. We also show some application to the regularity of paths for processes which take values in vector spaces. Finally we give a short proof of the Mendelson-Paouris result on sums of squares for empirical processes.
We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a method of Aida and Stroock for the classical logarithmic Sobolev inequality, we prove that if a measure on $\mathbb{R}^n$ satisfies a modified logarithmic Sobolev inequality then it satisfies a family of $L^p$-Sobolev-type inequalities with non-Euclidean norms of gradients (and dimension-independent constants). The latter are shown to yield various concentration-type estimates for deviations of smooth (not necessarily Lipschitz) functions and measures of enlargements of sets corresponding to non-Euclidean norms. We also prove a two-level concentration result for functions of bounded Hessian and measures satisfying the classical logarithmic Sobolev inequality.
We prove a law of large numbers for empirical approximations of the spectrum of a kernel integral operator by the spectrum of random matrices based on a sample drawn from a Markov chain, which complements the results by V. Koltchinskii and E. Giné for i.i.d. sequences. In a special case of Mercer's kernels and geometrically ergodic chains, we also provide exponential inequalities, quantifying the speed of convergence.
In this paper we study the regularity of paths in terms of properties of admissible nets. We show the right concentration inequality above the modulus of continuity. Using the approach we prove the Bernstein type inequality for the empirical processes. Therefore we obtain the best form of concentration for processes studied recently by Mendelson and Paouris. Results of this type are of importance in the compressed sensing theory.