Limit theorems for signatures

TL;DR

This work establishes strong invariance principles for normalized multiple iterated sums and integrals (signatures) of weakly dependent processes, showing that suitably rescaled iterated sums ${\mathbb S}_N^{(\nu)}$ can be coupled with Lyons extensions ${\mathbb W}_N^{(\nu)}$ built from Brownian motions with fixed covariance ${\varsigma}$. The authors derive explicit rate bounds in Prokhorov and Wasserstein metrics, first for $\nu=1,2$ via direct strong approximations and then for higher ranks using rough-path machinery and Chen relations, with extensions to discrete time, direct continuous time, and suspension (dynamical-system) setups. A comprehensive suite of auxiliary estimates (moments, Kolmogorov–Chentsov type results for multiplicative functionals, covariance-control, and strong approximation theorems) under weak dependence underpins the main results. The paper provides detailed variational- and supremum-norm bounds across all levels $\nu\le 4M$, enabling robust limit theorems for signatures in both probabilistic and dynamical-system contexts. Overall, the results advance rate-controlled weak-to-strong convergence for tensor-valued signatures, with potential applications in rough-path analysis and data-driven signature methods.

Abstract

We obtain strong moment invariance principles for normalized multiple iterated sums and integrals of the form $\mathbb{S}^{(ν)}(t)=N^{-ν/2}\sum_{0\leq k_1<...<k_ν\leq Nt}ξ(k_1)\otimes\cdots\otimesξ(k_ν)$, $t\in[0,T]$ and $\mathbb{S}_N^{(ν)}(t)=N^{-ν/2}\int_{0\leq s_1\leq...\leq s_ν\leq Nt}ξ(s_1)\otimes\cdots\otimesξ(s_ν)ds_1\cdots ds_ν$, where $\{ξ(k)\}_{-\infty<k<\infty}$ and $\{ξ(s)\}_{-\infty<s<\infty}$ are centered stationary vector processes with some weak dependence properties. We show, in particular, that (in both cases) the distribution of $\mathbb{S}^{(ν)}_N$ is $O(N^{-δ})$-close, $δ>0$ in the Prokhorov and the Wasserstein metrics to the distribution of certain stochastic processes $\mathbb{W}_N^{(ν)}$ constructed recursively starting from $W_N=\mathbb{W}_N^{(1)}$ which is a Brownian motion with covariances. This is done by constructing a coupling between $\mathbb{S}_N^{(1)}$ and $\mathbb{W}_N^{(1)}$, estimating directly the moment variational norm of $\mathbb{S}_N^{(ν)}-\mathbb{W}_N^{(ν)}$ for $ν=1,2$ and extending these estimates to $ν>2$ relying partially on arguments borrowed from the rough paths theory. In the continuous time we work both under direct weak dependence assumptions and also within the suspension setup which is more appropriate for applications in dynamical systems.

Theorems & Definitions (24)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof