RightsCreative Commons Attribution 4.0 International License. The theory is illustrated by concrete examples and an application to statistical lower bounds. DatesReceived: October 2017First available in Project Euclid: 27 July 2018, Permanent link to this documenthttps://projecteuclid.org/euclid.ejs/1532657104, Digital Object Identifierdoi:10.1214/18-EJS1456, Mathematical Reviews number (MathSciNet) MR3833470, Subjects Primary: 60G51: Processes with independent increments; Lévy processes 62M99: None of the above, but in this section Secondary: 60E07: Infinitely divisible distributions; stable distributions, KeywordsLévy processes Wasserstein distance total variation Toscani-Fourier distance statistical lower bound. Exponential ergodicity for Markov processes with random switching, Pareto Lévy measures and multivariate regular variation, Evolution of the Wasserstein distance between the marginals of two Markov processes, Two Moments Suffice for Poisson Approximations: The Chen-Stein Method, Probability measures, Lévy measures and analyticity in time, Stein’s method and Poisson process approximation for a class of Wasserstein metrics, Competing particle systems evolving by interacting Lévy processes, The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation, Functional quantization rate and mean regularity of processes with an application to Lévy processes, Poisson Process Approximations for the Ewens Sampling Formula. This turns out to be very useful in the context of Markov chains. 2, 2482--2514. doi:10.1214/18-EJS1456. A. Carrillo and G. Toscani: Contractive probability metrics and asymptotic behavior of dissipative kinetic equations.. [4] R. Cont and P. Tankov: Financial modelling with jump processes.. [5] C. G. Esseen: On mean central limit theorems.. [6] P. Étoré and E. Mariucci: L1-distance for additive processes with time-homogeneous Lévy measures.. [7] N. Fournier: Simulation and approximation of Lévy-driven stochastic differential equations.. [8] G. Gabetta, G. Toscani, and B. Wennberg: Metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation.. [9] J. Gairing, M. Högele, T. Kosenkova, and A. Kulik: Coupling distances between Lévy measures and applications to noise sensitivity of SDE.. [10] A. L. Gibbs and F. E. Su: On choosing and bounding probability metrics.. [11] C. R. Givens and R. M. Shortt: A class of Wasserstein metrics for probability distributions.. [12] B. V. Gnedenko and A. N. Kolmogorov: Limit distributions for sums of independent random variables.. [13] J. Jacod and M. Reiß: A remark on the rates of convergence for integrated volatility estimation in the presence of jumps.. [14] J. Jacod and A. N. Shiryaev: Limit theorems for stochastic processes. [16] F. Liese: Estimates of Hellinger integrals of infinitely divisible distributions.. [17] J. Mémin and A. N. Shiryayev: Distance de Hellinger-Kakutani des lois correspondant à deux processus à accroissements indépendants.. [18] M. H. Neumann and M. Reiß: Nonparametric estimation for Lévy processes from low-frequency observations.. [19] V. V. Petrov: Sums of independent random variables.. [20] E. Rio: Upper bounds for minimal distances in the central limit theorem.. [21] L. Rüschendorf and J. Woerner: Expansion of transition distributions of Lévy processes in small time.. [22] A. I. Sakhanenko: Estimates in an invariance principle.. [23] K. I. Sato: Lévy processes and infinitely divisible distributions.. [24] A. Source Electron. [15] Y. The second part of the questions asks about the coupling construction so that Pr (X ≠ Y) = T V (μ, ν) where X ∼ μ and Y ∼ ν. J. pling is also useful to bound the distance between probability measures. Mariucci, Ester; Reiß, Markus. J. The total variation distance between and (also called statistical distance) is the normalized ‘ 1-distance between the two probability measures: k k tv def= 1 2 X x2 [1] Y. Aït-Sahalia and J. Jacod: Testing for jumps in a discretely observed process.. [2] T. Bonis: Rates in the Central Limit Theorem and diffusion approximation via Stein’s Method.. [3] J. De nition 1. Connections to other metrics like Zolotarev and Toscani-Fourier distances are established. https://projecteuclid.org/euclid.ejs/1532657104, © Let and be two probability measures over a nite set . The theory is illustrated by concrete examples and an application to statistical lower bounds. Using the convolution structure, we further derive upper bounds for the total variation distance between the marginals of Lévy processes. Article information. The key insight from the coupling lemma is that the total variation distance between two distribu- tions and is bounded above by P(X6= Y) for any two random variables that are coupled with respect to and . Project Euclid, 60G51: Processes with independent increments; Lévy processes, 62M99: None of the above, but in this section, 60E07: Infinitely divisible distributions; stable distributions. Connections to other metrics like Zolotarev and Toscani-Fourier distances are established. Lindvall [10] explains how coupling was invented in the late 1930’s by Wolfgang Doeblin, and provides some historical context. 4.2.1 Bounding the total variation distance via coupling Let µ and ⌫ be probability measures on (S,S). A. Kutoyants: Statistical Inference for Spatial Poisson Processes. B. Tsybakov: Introduction to nonparametric estimation.. [25] C. Villani: Optimal transport: Old and New.. [26] V. M. Zolotarev: Modern theory of summation of random variables.. First, we need some de nitions and basic facts about statistical distance and couplings between proba-bility measures. Using the convolution structure, we further derive upper bounds for the total variation distance between the marginals of Lévy processes. Then the total variation distance will be 1 and no larger value can be achieved. Let µ and ⌫ be probability measures on (S,S).