arXiv subjects

David Kramer-Bang

Publications and source records attributed to David Kramer-Bang.

9 recordsLinked to original sources

arXiv2026

Generalised Random Parking for Trapeziums on a Strip

In this article, we study a generalisation of R\'{e}nyi's car-parking problem in which isosceles trapeziums are sequentially deposited on a strip. We derive an explicit formula for the parking constant in terms of the lengths of the two bases, recovering the classical rectangular and triangular models as special cases. We further obtain quantitative finite-size asymptotics for both the expected number and the variance of deposited trapeziums, with convergence rates that depend explicitly on the geometry of the deposited particle. In particular, although the recursive construction involves two different substrate geometries, their variances have the same leading asymptotic density. Finally, we show that the parking constant depends non-monotonically on the ratio of the two base lengths and possesses a unique minimiser, so that the least efficient shape is a genuine trapezium rather than a triangle.

math.PR↗

arXiv2026

Improved bounds on the inradius and inverse inradius process of the convex hull of planar Brownian motion

We study the inradius and inverse inradius of the convex hull of standard planar Brownian motion. Our main goals are to improve the existing bounds on the expected values of these quantities and to establish variance bounds. The main contributions of the paper, however, are the following intermediate results we obtain in order to compute our bounds. We find closed-form expressions for the expected values of the minimum and maximum of the ranges of two independent one-dimensional Brownian motions. Besides improving the upper bound on the expected inradius, this result also gives an explicit expression for a quantity that has been investigated repeatedly in the literature and for which only a numerical evaluation has been known so far. Furthermore, we study the exit time of planar Brownian motion from an unbounded region that we call the hourglass domain and compute its first two moments. This result leads to a nearly fivefold improvement of the previous upper bound on the expected inverse inradius and plays an essential role in establishing the corresponding upper bound on the variance.

math.PR↗

arXiv2025

Multivariate CLT for L\'evy processes: convergence rates without moment assumptions

We prove that the norm of a $d$-dimensional L\'evy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time $t$ and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure $t^{-1} dt$ on $[1,\infty)$. We further prove that under the standard $\sqrt{t}$-scaling, the corresponding convex distance is integrable if and only if the norm of the L\'evy process has a finite $(2+\log)$-moment. Both equivalences also hold for the integrability with respect to $t^{-1} dt$ of the multivariate Kolmogorov distance. Our results imply: (I) polynomial Berry-Esseen bounds on the rate of convergence in the convex distance in the CLT for L\'evy processes cannot hold without finiteness of $(2+\delta)$-moments for some $\delta>0$ and (II) integrability of the convex distance with respect to $t^{-1} dt$ in the domain of non-normal attraction cannot occur for any scaling function.

math.PR↗

arXiv2025

Rates of convergence for multivariate SDEs driven by L\'evy processes in the small-time stable domain of attraction

This paper explores the rates of convergence of solutions for multivariate stochastic differential equations (SDEs) driven by L\'evy processes within the small-time stable domain of attraction (DoA). Explicit bounds are derived for the uniform Wasserstein distance between solutions of two L\'evy-driven SDEs, expressed in terms of driver characteristics. These bounds establish convergence rates in probability for drivers in the DoA, and yield uniform Wasserstein distance convergence for SDEs with additive noise. The methodology uses two couplings for L\'evy driver jump components, leading to sharp convergence rates tied to the processes' intrinsic properties.

math.PR↗

arXiv2025

Quantitative bounds for high-dimensional non-linear functionals of Gaussian processes

In this paper, we establish explicit quantitative Berry-Esseen bounds in the hyper-rectangle distance $d_R$, the convex distance $d_{\mathscr{C}}$ and the $1$-Wasserstein distance $d_W$ for high-dimensional, non-linear functionals of Gaussian processes, allowing for strong dependence between variables. Our main result demonstrates that, under a smoothness assumption, the convergence rate under $d_R$ is sub-polynomial in the dimension and polynomial under $d_{\mathscr{C}}$ and $d_W$. To the best of our knowledge, our results under $d_R$ provide the first explicit sub-polynomial bound for high-dimensional, non-linear functionals of Gaussian processes beyond the i.i.d. setting. Building on this, we derive explicit Berry-Esseen bounds under both $d_R$ and $d_{\mathscr{C}}$ for multiple statistical examples, such as the method of moments, empirical characteristic functions, empirical moment-generating functions, and functional limit theorems in high-dimensional settings.

math.PR↗

arXiv2025

Exponential dimensional dependence in high-dimensional Hermite method of moments

It is numerically well known that moment-based tests for Gaussianity and estimators become increasingly unreliable at higher moment orders; however, this phenomenon has lacked rigorous mathematical justification. In this work, we establish quantitative bounds for Hermite-based moment tests, with matching exponential upper and lower bounds. Our results show that, even under ideal conditions with i.i.d. standard normal data, the sample size must grow exponentially with the highest moment order $d$ used in the test. These bounds, derived under both the convex distance and the Kolmogorov-Smirnov distance, are applied to classical procedures, such as the Shenton-Bowman test.

math.ST↗

arXiv2025

Fourth-Moment Theorems for Sums of Multiple Integrals

Nualart & Pecatti ([Nualart and Peccati, 2005, Thm 1]) established the first fourth-moment theorem for random variables in a fixed Wiener chaos, i.e. they showed that convergence of the sequence of fourth moments to the fourth moment of the standard Gaussian distribution is sufficient for weak convergence to the standard Gaussian. In this paper, we provide what we believe to be the first generalization to chaos expansions with more than a single term. Specifically, we show that a fourth-moment theorem holds for random variables consisting of sums of two multiple integrals of orders $p, q \in N$, where $p, q$ have different parities. Furthermore, we show that such random variables cannot themselves be Gaussian, again generalizing what is known for the fixed Wiener chaos setting. Finally, we show a fourth-moment theorem for variables with infinite Wiener chaos expansions when the terms in the expansions are independent and satisfy an additional regularity condition in terms of the Ornstein-Uhlenbeck operator.

math.PR↗

arXiv2024

Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction

We develop two novel couplings between general pure-jump L\'evy processes in $\R^d$ and apply them to obtain upper bounds on the rate of convergence in an appropriate Wasserstein distance on the path space for a wide class of L\'evy processes attracted to a multidimensional stable process in the small-time regime. We also establish general lower bounds based on certain universal properties of slowly varying functions and the relationship between the Wasserstein and Toscani--Fourier distances of the marginals. Our upper and lower bounds typically have matching rates. In particular, the rate of convergence is polynomial for the domain of normal attraction and slower than a slowly varying function for the domain of non-normal attraction.

math.PR↗

arXiv2022

H\"older continuity of the convex minorant of a L\'evy process

We characterise the H\"older continuity of the convex minorant of most L\'evy processes. The proof is based on a novel connection between the path properties of the L\'evy process at zero and the boundedness of the set of $r$-slopes of the convex minorant.

math.PR↗