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May 2021 Total variation distance for discretely observed Lévy processes: A Gaussian approximation of the small jumps
Alexandra Carpentier, Céline Duval, Ester Mariucci
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Ann. Inst. H. Poincaré Probab. Statist. 57(2): 901-939 (May 2021). DOI: 10.1214/20-AIHP1102

Abstract

It is common practice to treat small jumps of Lévy processes as Wiener noise and to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation; if the total variation distance between two statistical models converges to zero, then no test can be constructed to distinguish the two models and they are therefore asymptotically equally informative. We elaborate a fine analysis of a Gaussian approximation for the small jumps of Lévy processes in total variation distance. Non-asymptotic bounds for the total variation distance between n discrete observations of small jumps of a Lévy process and the corresponding Gaussian distribution are presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of Lévy processes are provided. The theory is illustrated by concrete examples.

Il est habituel d’assimiler les petits sauts d’un processus de Lévy à un mouvement brownien et d’approcher leurs marginales par des distributions gaussiennes. Cependant, les résultats permettant de quantifier cette approximation selon une métrique donnée sont rares. Dans cet article, nous la quantifions pour la distance en variation totale. Un tel choix s’explique par son interprétation statistique : si la distance en variation totale entre deux modèles statistiques tend vers 0, alors il n’existe aucun test permettant de distinguer les deux modèles, qui sont alors asymptotiquement équivalents. Nous contrôlons ici finement la distance en variation totale entre n incréments des petits sauts d’un processus de Lévy et n variables aléatoires gaussiennes : des bornes non asymptotiques pour la distance en variation totale sont données et discutées. Une conséquence de ces résultats est l’obtention de nouvelles bornes supérieures pour le contrôle en variation totale entre n incréments de deux processus de Lévy. Plusieurs exemples viennent illustrer ces résultats.

Acknowledgements

The work of A. Carpentier is partially supported by the Deutsche Forschungsgemeinschaft (DFG) Emmy Noether grant MuSyAD (CA 1488/1-1), by the DFG – 314838170, GRK 2297 MathCoRe, by the DFG GRK 2433 DAEDALUS, by the DFG CRC 1294 ‘Data Assimilation’, Project A03, and by the UFA-DFH through the French–German Doktorandenkolleg CDFA 01-18.

The work of E. Mariucci has been partially funded by the Federal Ministry for Education and Research through the Sponsorship provided by the Alexander von Humboldt Foundation, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 314838170, GRK 2297 MathCoRe, and by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1294 ‘Data Assimilation’.

Citation

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Alexandra Carpentier. Céline Duval. Ester Mariucci. "Total variation distance for discretely observed Lévy processes: A Gaussian approximation of the small jumps." Ann. Inst. H. Poincaré Probab. Statist. 57 (2) 901 - 939, May 2021. https://doi.org/10.1214/20-AIHP1102

Information

Received: 21 May 2019; Revised: 18 March 2020; Accepted: 1 September 2020; Published: May 2021
First available in Project Euclid: 13 May 2021

Digital Object Identifier: 10.1214/20-AIHP1102

Subjects:
Primary: 60G51, 62M99
Secondary: 60E99

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 2 • May 2021
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