Extremes of Gaussian random fields
The tail behaviour of suprema, infima and sojourn times of Gaussian processes and fields, scalar and vector-valued: how a rough field exceeds a high level, how long it stays there, and the Pickands–Piterbarg-type constants that quantify it. Recent work extends this to vector-valued stationary fields and to the much harder question of infima — a field staying high everywhere on a set.
Sojourns of vector-valued fields · Infima of Gaussian processes · illustration: Euler characteristic vs Pickands · illustration: staying high everywhere
Shift-invariant and max-stable structures
Stationarity in unusual places: max-stable fields indexed by rooted trees that look the same from every branch, Poisson particle systems that are stationary although each particle is anchored at its own origin, and the persistence of Brown–Resnick processes. The common thread is a representation of the asymptotic constants through shift-invariant measures.
Max-stable fields on rooted trees · Stationary Poisson particle systems · illustration: a max-stable field on a binary tree · illustration: a particle system with no clock of its own
Regular variation and tail measures
Regular variation of random processes and fields on general Polish spaces, and the tail measures that describe their extremes. Tail measures can be defined through a countable family of homogeneous coordinate maps, which makes regular variation of vector-valued càdlàg processes indexed by ℝl as tractable as the stationary time-series case. In the multiradial version each component of a field has its own threshold growing at its own rate; the limit measure is then homogeneous in every component separately, and every shift-invariant tail measure of this kind is the tail measure of a strictly stationary field. Underneath sits a theory of multihomogeneous measures — measures that scale under the action of (0,∞)q — with stochastic polar representations, canonical representers and tail-overlap kernels.
Tail measures and regular variation (EJP, with M. Bladt and G. Shevchenko) · Multiradial regular variation · Multihomogeneous measures and stochastic polar representations
Ruin and rare events for Brownian risk models
Finite-horizon ruin in forms that matter for reporting and regulation — ruin in every block or every sliding window, simultaneous ruin of several lines of business on a metric tree, Parisian ruin of branching populations — and Brownian motion with exponential resetting, where restarts make the search for a high level efficient. In each case the rare event is driven by a single mechanism that the asymptotics make explicit.
Block and sliding-window ruin · Ruin on Brownian metric trees · Parisian ruin of branching Brownian motion · Brownian motion with resetting · illustration: ruin in every block · illustration: back to square one
Laplace principles and exponential functionals
Tail and density asymptotics for homogeneous functionals of Gaussian vectors and for exponential integrals of Gaussian processes, via a radial–angular Laplace principle: the rare event chooses a direction, and the price of the constraint is read off from the geometry of a switching curve or an RKHS minimiser. Closely related: the long memory of Gaussian queues, whose workload covariance decays like a power of the lag.
Radial–angular Laplace principle · Gaussian exponential integrals · Covariance decay for Gaussian queues · illustration: the angle a rare event chooses · illustration: the most likely path
Long-standing collaborators
Krzysztof Dębicki, Zbigniew Michna, Georgiy Shevchenko, and my current and former students, among many others.