A distance based measure of dependence is proposed for stable distributions that completely characterizes independence for a bivariate stable distribution. Properties of this measure are analyzed, and contrasted with the covariation and co-difference. A sample analog of the measure is defined and demonstrated on simulated and real data, including time series and distributions in the domain of attraction of a stable law.
This is joint work with Tuncay Alparslan.[-]

Consider random conductances that allow long range jumps. In particular we consider conductances $C_{xy} = w_{xy}|x − y|^{−d−\alpha}$ for distinct $x, y \in Z^d$ and $0 < \alpha < 2$, where $\lbrace w_{xy} = w_{yx} : x, y \in Z^d\rbrace$ are non-negative independent random variables with mean 1. We prove that under some moment conditions for $w$, suitably rescaled Markov chains among the random conductances converge to a rotationally symmetric $\alpha$-stable process almost surely w.r.t. the randomness of the environments. The proof is a combination of analytic and probabilistic methods based on the recently established de Giorgi-Nash-Moser theory for processes with long range jumps. If time permits, we also discuss quenched heat kernel estimates as well. This is a joint work with Xin Chen (Shanghai) and Jian Wang (Fuzhou).[-]

We study the convergence of systems of interacting particles driven by Poisson random measure, having mean field interactions and position dependent jump rate. Jumps are simultaneous, that is, at each jump time, all particles of the system are affected by this jump and receive a positive random jump height. This random kick is distributed according to a one-sided alpha-stable law and scaled in $N^{-1/\infty }$, where $N$ is the size of the system. This particular scaling implies that the limit of the empirical measures of the system is random, describing the conditional distribution of one particle in the limit system. Such limits are conditional McKean-Vlasov limits. The conditioning in the limit measure reflects the dependencies between coexisting particles in the limit system such that we are dealing with a conditional propagation of chaos property. I will spend some time to explain the explicit structure of the limit system which turns out to be the solution of a non-linear SDE driven by Poisson random measure and an independent alpha-stable subordinator. In a second part of the talk I discuss strong error bounds allowing us to control the rate of convergence of the finite particle system to the limit system.
This is a joint work with Dasha Loukianova (Université d'Evry).[-]