Exact Signature Tail Asymptotics for Pure Rough Paths

We prove~\cite[Conjecture 2.12]{BGS20} on the signature tail asymptotics of pure rough paths and extend it to arbitrary reasonable tensor norms. In more details, let $$\mathbf X_t=\exp(tl) \,\text{ with }\, l=l_1+\cdots+l_m\,\text{ and }\, l_r\in\mathcal L_r(V),$$ be a pure $m$-rough path over a finite dimensional real or complex Banach space, and equip the tensor powers of $V$ with arbitrary reasonable tensor algebra norms. We prove that $$\limsup_{n\to\infty}\left(\left(\frac{n}{m}\right)!\left\|\pi_n(\exp l)\right\|_n\right)^{m/n}=\|l_m\|_m .$$ In particular, this identifies the signature tail with the local $m$-variation of the pure rough path. The upper bound was obtained in~\cite{BGS20}; the main contribution of the paper is the matching lower bound. Its proof is based on finite dimensional developments and a norming cyclic construction. For every top-level tensor $l_m$, we also build a contractive development in which $\|l_m\|_m$ appears as an eigenvalue at degree $m$.