Related papers: How close are time series to power tail L\'evy dif…
We study the empirical version of halfspace depths with the objective of establishing a connection between the rates of convergence and the tail behaviour of the corresponding underlying distributions. The intricate interplay between the…
Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy…
Extreme events are often multivariate in nature. A compound extreme occurs when a combination of variables jointly produces a significant impact, even if individual components are not necessarily marginally extreme. Compound extremes have…
A jumping process, defined in terms of jump size distribution and waiting time distribution, is presented. The jumping rate depends on the process value. The process, which is Markovian and stationary, relaxes to an equilibrium and is…
L\'{e}vy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the L\'{e}vy walk with the exponent of the power-law distributed flight time $\alpha\in(0,2)$. We…
Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of L\'evy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker,…
We present a novel idea for a coupling of solutions of stochastic differential equations driven by L\'{e}vy noise, inspired by some results from the optimal transportation theory. Then we use this coupling to obtain exponential…
We say that a random variable is $light$-$tailed$ if moments of order $2+\epsilon$ are finite for some $\epsilon>0$; otherwise, we say that it is $heavy$-$tailed$. We study queueing networks that operate under the Max-Weight scheduling…
In the Heliosphere, power-law particle distributions are observed e.g. upstream of interplanetary shocks, which can result from superdiffusive transport. This non-Gaussian transport regime may result from intermittent magnetic field…
Large deviation principles and related results are given for a class of Markov chains associated to the "leaves" in random recursive trees and preferential attachment random graphs, as well as the "cherries" in Yule trees. In particular,…
We prove a version of Nagaev's theorem for the branching random walk with heavy-tailed associated random walk. For a branching random walk on $\mathbb{R}$ we consider the random measure $Z_n = \sum_{|u|=n} e^{-V_u} \delta_{V_u}$ where…
In classical diffusion, particle step-sizes have a Gaussian distribution. However, in superdiffusion, they have power-law tails, with transport dominated by rare, long L\'evy flights. Similarly, if the time interval between scattering…
Among Markovian processes, the hallmark of L\'evy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that L\'evy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that…
We extend the L\'evy Langevin Monte Carlo method studied by Oechsler in 2024 to the setting of a target distribution with heavy tails: Choosing a target distribution from the class of subexponential distributions we prove convergence of a…
Consider a Markov process \omega_t at equilibrium and some event C (a subset of the state-space of the process). A natural measure of correlations in the process is the pairwise correlation \Pr[\omega_0,\omega_t \in C] - \Pr[\omega_0 \in…
Modeling uncertainty in heavy-tailed time series remains a critical challenge for deep probabilistic forecasting models, which often struggle to capture abrupt, extreme events. While L\'evy stable distributions offer a natural framework for…
We prove the well-posedness of solutions to McKean-Vlasov stochastic differential equations driven by L\'evy noise under mild assumptions where, in particular, the L\'evy measure is not required to be finite. The drift, diffusion and jump…
We analyze confining mechanisms for L\'{e}vy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump - type processes are…
Stable laws can be tempered by modifying the L\'evy measure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops random walk…
The big jump principle explains the emergence of extreme events for physical quantities modelled by a sum of independent and identically distributed random variables which are heavy-tailed. Extreme events are large values of the sum and…