Bregman variation of semimartingales
We define Bregman variation of semimartingales. We give its pathwise representation, It\^o-type isometry for martingales, and applications to harmonic analysis.
math.PR↗
arXiv subjects
Dominik Kutek
Publications and source records attributed to Dominik Kutek.
Exponential inequalities and laws of the iterated logarithm for multiple Poisson--Wiener integrals and Poisson $U$-statistics
We prove tail and moment inequalities for multiple stochastic integrals on the Poisson space and for Poisson $U$-statistics. We use them to demonstrate the Law of the Iterated Logarithm for these processes when the intensity of the Poisson process tends to infinity, with normalization depending on the degree of the multiple stochastic integral or degeneracy of the kernel defining the $U$-statistic. We apply our results to several classical functionals of Poisson point processes, obtaining improvements or complements of known concentration of measure results as well as new laws of the iterated logarithm. Examples include subgraph counts and power length functionals of geometric random graphs, intersections of Poisson $k$-flats, quadratic functionals of the Ornstein--Uhlenbeck L\'evy process and $U$-statistics of marked processes. Keywords: Poisson point process, $U$-statistics, multiple stochastic integrals, concentration of measure, The Law of the Iterated Logarithm
math.PR↗
On Orlicz spaces satisfying the Hoffmann-Jørgensen inequality
Building on Talagrand's proof of the Hoffmann-Jørgensen inequality for $L_p$ spaces and its version for the exponential Orlicz spaces we provide a full characterization of Orlicz functions $Ψ$ for which an analogous inequality holds in the Orlicz space $L_Ψ(F)$, where $F$ is an arbitrary Banach space. As an application we present a characterization of Talagrand-type concentration inequality for suprema of empirical processes with envelope in $L_Ψ$ (equivalently for sums of independent $F$-valued random variables in $L_Ψ(F)$). This result generalizes in particular an inequality by the first-named author concerning exponentially integrable summands and a recent inequality due to Chamakh-Gobet-Liu on summands with $β$-heavy tails. Another corollary concerns concentration for convex functions of independent, unbounded random variables, generalizing recent results due to Klochkov-Zhivotovskiy and Sambale. We also obtain a corollary concerning boundedness in $L_Ψ(F)$ of partial sums of a series of independent random variables, generalizing the original result by Hoffmann-Jørgensen.
math.PR↗