Lindvall [10] explains how coupling was invented in the late 1930’s by Wolfgang Doeblin, and provides some historical context. 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. Using the convolution structure, we further derive upper bounds for the total variation distance between the marginals of Lévy processes. [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. The symbol tvis used for the total variation distance, which is deﬁned at the beginning of Chapter 2. https://projecteuclid.org/euclid.ejs/1532657104, © Project Euclid, 60G51: Processes with independent increments; Lévy processes, 62M99: None of the above, but in this section, 60E07: Infinitely divisible distributions; stable distributions. Electron. The theory is illustrated by concrete examples and an application to statistical lower bounds. Source Electron. De nition 1. J. Statist., Volume 12, Number 2 (2018), 2482-2514. 4.2.1 Bounding the total variation distance via coupling Let µ and ⌫ be probability measures on (S,S). RightsCreative Commons Attribution 4.0 International License. Mariucci, Ester; Reiß, Markus. Article information. 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. This turns out to be very useful in the context of Markov chains. Let µ and ⌫ be probability measures on (S,S). 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 . Recall the deﬁnition of the total variation distance kµ⌫k TV:= sup A2S |µ(A)⌫(A)|. [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. 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 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. J. Statist. Standard references for coupling are Lindvall [11] and pling is also useful to bound the distance between probability measures.

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