Embedded Markov chain approximations in Skorokhod topologies

In order to approximate a continuous time stochastic process by discrete time Markov chains one has several options to embed the Markov chains into continuous time processes. On the one hand there is the Markov embedding, which uses exponential waiting times. On the other hand each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for $J_1$, the linear interpolation embedding for $M_1$, the multi step embedding for $J_2$ and a more general embedding for $M_2$. We show that the convergence of the step function embedding in $J_1$ implies the convergence of the other embeddings in the corresponding topologies, respectively. For the converse statement a $J_1$-tightness condition for embedded Markov chains is given. The result relies on various representations of the Skorokhod topologies. Additionally it is shown that $J_1$ convergence is equivalent to the joint convergence in $M_1$ and $J_2$.Comment: To appear in Probability and Mathematical Statistic

The Euler scheme for Feller processes

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented. In particular the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of L\'evy processes and thus the Euler scheme can be used for simulation by applying standard techniques from L\'evy processes

Constructions of Coupling Processes for L\'evy Processes

We construct optimal Markov couplings of L\'{e}vy processes, whose L\'evy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.Comment: 16 page

Detecting independence of random vectors: generalized distance covariance and Gaussian covariance

Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Sz\'{e}kely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric L\'{e}vy measures and the corresponding real-valued continuous negative definite functions. The L\'{e}vy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework. From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance Multivariance: New dependence measures for random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the present article.Comment: Published at https://doi.org/10.15559/18-VMSTA116 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/