Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length

Abstract

For a path of length L>0, if for all n≥1, we multiply the n-th term of the signature by n!L−n, 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 approaches infinity.

Citation information

Ni, H; Lyons, T; Chang, J; (2018) Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length. Comptes Rendus Mathématique 10.1016/j.crma.2018.05.010

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