We construct optimal Markov couplings of Lévy processes, whose Lévy (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.
We introduce two new measures for the dependence of n ≥ 2 random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted L 2 -distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to n-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of n random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives. MSC classification: 62H20; 60E10, 62G10, 62G15, 62G20
Key words Pseudo-differential operator, Markov process, fundamental solution, parametrix for parabolic equation
MSC (2000) 35S10, 60J35, 47D07We use the method proposed by H. Kumano-go in the classical case to construct a parametrix of the equation On the one hand we know that e −tq(x,D) is a pseudo-differential operator, but we don't know its symbol.On the other hand it is known that e −tq (x,D) u is a solution to the equationH. Kumanogo constructed an approximation for the symbol of the solution operator to this equation. In his case the operator q(x, D) had a classical symbol, i.e., it was in the class S m ρ,δ . We prove an analogous result for pseudo-differential operators with symbols in W. Hoh's class. This approximation is then also an approximation of the symbol of the Feller semigroup generated by −q(x, D). In particular we will get σ (T t ) (x, ξ) = e −tq(x,ξ) + r 0 (t; x, ξ) ,where the remainder term r 0 satisfies some smallness conditions. *
We consider Feller processes whose generators have the test functions as an operator core. In this case, the generator is a pseudo differential operator with negative definite symbol q(x, ξ). If |q(x, ξ)| < c(1 + |ξ| 2 ), the corresponding Feller process can be approximated by Markov chains whose steps are increments of Lévy processes.This approximation can easily be used for a simulation of the sample path of a Feller process.Further, we provide conditions in terms of the symbol for the transition operators of the Markov chains to be Feller. This gives rise to a sequence of Feller processes approximating the given Feller process.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.