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Rafał Martynek

Publications and source records attributed to Rafał Martynek.

7 recordsLinked to original sources
arXiv2026

Operator-norm Sudakov minoration for Gaussian chaos of order two

We prove that an operator-norm separated family of matrices satisfies $\mathbb{E}\sup_{A\in T} G^{T}AG' \geq ca\log |T|$, where G,G' are independent standard Gaussian vectors and a is the separation. The main information estimate concerns arbitrary separated coisometries: conditional entropy is bounded by a source-dependent operator energy times $\log|T|$, up to an additive quadratic term in the common row dimension. An adaptive Gaussian experiment proves this estimate by charging actual information increments to one weighted posterior-entropy potential. Convex separation and a Gaussian covering estimate then yield a bounded-radius result. To reach the general case, we first choose an operator scale preserving the Sudakov ratio, apply the known Hilbert-Schmidt minoration, and recompute a common Gaussian block compression at the retained entropy. This ordering preserves the normalization needed by the coisometry argument.

math.PR↗
arXiv2026

Small cover approach to the suprema of positive canonical processes

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.

math.PR↗
arXiv2024

The suprema of infinitely divisible processes

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.

math.PR↗
arXiv2020

On tails of symmetric and totally asymmetric $α$-stable distributions

We estimate up to universal constants tails of symmetric and totally asymmetric 1-dimensional $α$-stable distributions in terms of functions of the parameters of these distributions. In particular, for values of $α$ close to $2$ we specify where exactly the tail changes from being Gaussian and starts to behave like in the Pareto distribution

math.PR↗
arXiv2019

A Lévy-Ottaviani type inequality for the Bernoulli process on an interval

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 $C=2$.

math.PR↗
arXiv2019

On a contraction property of Bernoulli canonical processes

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 $I\subset\mathbb{N}$, $I^c=\mathbb{N}\backslash I$.

math.PR↗