arXiv subjects

Cees de Valk

Publications and source records attributed to Cees de Valk.

5 recordsLinked to original sources

arXiv2026

Zero-couplings of infinite measures with cyclically monotone support and multivariate regular variation

We study cyclically monotone transport plans between measures in $\mathrm{M}_0(\mathbb{R}^d)$, the class of Borel measures on $\mathbb{R}^d \setminus \{0\}$ that are finite on sets bounded away from the origin but may have infinite total mass. We avoid moment assumptions and allow the transport cost to be infinite. This framework naturally arises for exponent measures in multivariate regular variation and includes other examples such as L\'evy measures. We introduce the notion of a zero-coupling and establish existence of cyclically monotone zero-couplings for arbitrary pairs of measures in $\mathrm{M}_0(\mathbb{R}^d)$. Under a Hausdorff-dimension condition on the first measure and when at least one of the two measures has infinite mass, we prove uniqueness of the cyclically monotone zero-coupling, yielding an analogue of the Brenier--McCann theorem in this infinite-measure setting. We further derive a representation of such couplings through gradients of closed convex functions and identify conditions under which the zero-coupling is proper in the sense that the second measure is equal to the restriction to the punctured space of the push-forward of the first measure by a cyclically monotone transport map. Finally, we apply these results to regularly varying probability measures. We show that a cyclically monotone coupling between two such distributions admits a tail limit that coincides with the unique proper cyclically monotone zero-coupling between the corresponding exponent measures.

math.PR↗

arXiv2025

An extreme value method to study decadal hurricane wind trends

This paper presents a method developed using techniques from extreme value theory to estimate smooth wind-speed percentiles, allowing us to consider more extreme wind speeds while being less sensitive to the noise that stems from the scarcity of extreme data. A reliable characterisation of wind extremes is the first required step for studying decadal trends in tropical-cyclone and extra-tropical-cyclone winds. We develop a percentile-smoothing method using ASCAT-A Level-3 products, focusing on a number of tropical basins (Caribbean and Atlantic), estimate the uncertainty with the block-bootstrap technique to address the issue of dependency, and apply our method to both scatterometer winds (ASCAT-A at two different resolutions) and collocated ERA5 model data. The results obtained are very robust at basin level, without having to rely on a strong assumption for the distribution tail: they are very consistent whether we use exponential fits, generalised-Pareto fits, or even no fit at all, down to at least truly extreme wind percentiles such as 99.999th (main result), and remain quite consistent within uncertainties down to 99.9999th. As ensuring scientifically sound decadal-trend conclusions would require going back sufficiently in time, spanning the lifetimes of different instruments with different characteristics and extreme-wind statistics, a natural follow-on study would be to apply this method not only to ASCAT, but also to its predecessors on QuikSCAT and ERS - comparing each scatterometer individually against ERA5, and also to each other as partial overlaps exist between instruments.

physics.ao-ph↗

arXiv2019

Tails of optimal transport plans for regularly varying probability measures

For the basic case of $L_2$ optimal transport between two probability measures on a Euclidean space, the regularity of the coupling measure and the transport map in the tail regions of these measures is studied. For this purpose, Robert McCann's classical existence and uniqueness results are extended to a class of possibly infinite measures, finite outside neighbourhoods of the origin. For convergent sequences of pairs of such measures, the stability of the multivalued transport maps is considered, and a useful notion of locally uniform convergence of these maps is verified under light assumptions. Applied to regularly varying probability measures, these general results imply the existence of tail limits of the transport plan and the coupling measure, these objects exhibiting distinct types of homogeneity.

math.PR↗

arXiv2016

Approximation of high quantiles from intermediate quantiles

Motivated by applications requiring quantile estimates for very small probabilities of exceedance, this article addresses estimation of high quantiles for probabilities bounded by powers of sample size with exponents below -1. As regularity assumption, an alternative to the Generalised Pareto tail limit is explored for this purpose. Motivation for the alternative regularity assumption is provided, and it is shown to be equivalent to a limit relation for the logarithm of survival function, the log-GW tail limit, which generalises the GW (Generalised Weibull) tail limit, a generalisation of the Weibull tail limit. The domain of attraction is described, and convergence results are presented for quantile approximation and for a simple quantile estimator based on the log-GW tail. Simulations are presented, and advantages and limitations of log-GW-based estimation of high quantiles are indicated.

math.ST↗

arXiv2015

Approximation and estimation of very small probabilities of multivariate extreme events

This article discusses modelling of the tail of a multivariate distribution function by means of a large deviation principle (LDP), and its application to the estimation of the probability of a multivariate extreme event from a sample of n iid random vectors, with the probability bounded by powers of sample size with exponents below -1. One way to view classical tail limits is as limits of probability ratios. In contrast, the tail LDP provides asymptotic bounds or limits for log-probability ratios. After standardising the marginals to standard exponential, dependence is represented by a homogeneous rate function. Furthermore, the tail LDP can be extended to represent both dependence and marginals, the latter implying marginal log-GW tail limits. A connection is established between the tail LDP and residual tail dependence (or hidden regular variation) and a recent extension of it. Under a smoothness assumption, they are implied by the tail LDP. Based on the tail LDP, a simple estimator for very small probabilities of extreme events is formulated. It avoids estimation of the rate function by making use of its homogeneity. Strong consistency in the sense of convergence of log-probability ratios is proven. Simulations and an application illustrate the difference between the classical approach and the LDP-based approach.

math.ST↗