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Comptes Rendus Mathématique
Volume 357, n° 2
pages 120-129 (février 2019)
Doi : 10.1016/j.crma.2018.12.006
Received : 17 August 2018 ;  accepted : 21 December 2018
A non-vanishing property for the signature of a path
Une propriété de non-nullité pour la signature d'un chemin
 

Horatio Boedihardjo a , Xi Geng b
a Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom 
b Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States 

Abstract

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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Résumé

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.

The full text of this article is available in PDF format.
1  Indeed, in [[3]] the authors claimed the convergence as   without further restrictions. However, a careful examination of the proof suggests that the convergence was only proved along degrees at which the signature is nonzero. This was corrected in the corrigendum [[4]] of [[3]]. Theorem 1 stated above is the corrected version.


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