[Supermultiplicativité et une borne inférieure pour la décroissance de la signature d'un chemin de longueur finie]
Pour une trajectoire de longueur , si l'on multiplie le n-ième terme de la signature par pour tout , la signature ainsi obtenue est dite « normalisée ». Il a été établi (T. J. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths, Springer, 2007) que la norme du n-ième terme de la signature normalisée d'une trajectoire à variation bornée est majorée par 1. Dans cet article, nous étudions la super-multiplicativité de la norme de la signature d'une trajectoire de longueur finie, et nous démontrons, à l'aide du lemme de Fekete, l'existence d'une limite non nulle lorsque n tend l'infini pour la racine n-ième de la norme du n-ième terme de la signature normalisée.
For a path of length , if for all , we multiply the n-th term of the signature by , we say that the resulting signature is ‘normalised’. It has been established (T. J. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths, Springer, 2007) that the norm of the n-th term of the normalised signature of a bounded-variation path is bounded above by 1. In this article, we discuss the super-multiplicativity of the norm of the signature of a path with finite length, and prove by Fekete's lemma the existence of a non-zero limit of the n-th root of the norm of the n-th term in the normalised signature as n approaches infinity.
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@article{CRMATH_2018__356_7_720_0,
author = {Chang, Jiawei and Lyons, Terry and Ni, Hao},
title = {Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length},
journal = {Comptes Rendus. Math\'ematique},
pages = {720--724},
publisher = {Elsevier},
volume = {356},
number = {7},
year = {2018},
doi = {10.1016/j.crma.2018.05.010},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2018.05.010/}
}
TY - JOUR AU - Chang, Jiawei AU - Lyons, Terry AU - Ni, Hao TI - Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length JO - Comptes Rendus. Mathématique PY - 2018 SP - 720 EP - 724 VL - 356 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2018.05.010/ DO - 10.1016/j.crma.2018.05.010 LA - en ID - CRMATH_2018__356_7_720_0 ER -
%0 Journal Article %A Chang, Jiawei %A Lyons, Terry %A Ni, Hao %T Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length %J Comptes Rendus. Mathématique %D 2018 %P 720-724 %V 356 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2018.05.010/ %R 10.1016/j.crma.2018.05.010 %G en %F CRMATH_2018__356_7_720_0
Chang, Jiawei; Lyons, Terry; Ni, Hao. Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length. Comptes Rendus. Mathématique, Tome 356 (2018) no. 7, pp. 720-724. doi : 10.1016/j.crma.2018.05.010. http://www.numdam.org/articles/10.1016/j.crma.2018.05.010/
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