[Une propriété de non-nullité pour la signature d'un chemin]
Nous montrons que la signature d'un chemin continu, de longueur finie, dans un espace de Banach réel, ne peut pas avoir une infinité de composantes nulles, à moins d'être de type arbre. En particulier, cela nous permet de renforcer un théorème limite pour la signature, récemment obtenu par Chang, Lyons et Ni. Notre démonstration repose sur un argument de complexification et des résultats profonds d'approximations polynomiales holomorphes de la théorie de plusieurs variables complexes.
We prove that a continuous path with finite length in a real Banach space cannot have infinitely many zero components in its signature unless it is tree-like. In particular, this allows us to strengthen a limit theorem for signature recently proved by Chang, Lyons, and Ni. What lies at the heart of our proof is a complexification idea together with deep results from holomorphic polynomial approximations in the theory of several complex variables.
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@article{CRMATH_2019__357_2_120_0,
author = {Boedihardjo, Horatio and Geng, Xi},
title = {A non-vanishing property for the signature of a path},
journal = {Comptes Rendus. Math\'ematique},
pages = {120--129},
publisher = {Elsevier},
volume = {357},
number = {2},
year = {2019},
doi = {10.1016/j.crma.2018.12.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2018.12.006/}
}
TY - JOUR AU - Boedihardjo, Horatio AU - Geng, Xi TI - A non-vanishing property for the signature of a path JO - Comptes Rendus. Mathématique PY - 2019 SP - 120 EP - 129 VL - 357 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2018.12.006/ DO - 10.1016/j.crma.2018.12.006 LA - en ID - CRMATH_2019__357_2_120_0 ER -
%0 Journal Article %A Boedihardjo, Horatio %A Geng, Xi %T A non-vanishing property for the signature of a path %J Comptes Rendus. Mathématique %D 2019 %P 120-129 %V 357 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2018.12.006/ %R 10.1016/j.crma.2018.12.006 %G en %F CRMATH_2019__357_2_120_0
Boedihardjo, Horatio; Geng, Xi. A non-vanishing property for the signature of a path. Comptes Rendus. Mathématique, Tome 357 (2019) no. 2, pp. 120-129. doi : 10.1016/j.crma.2018.12.006. http://www.numdam.org/articles/10.1016/j.crma.2018.12.006/
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