Limit theorems for signatures

Abstract

We obtain strong moment invariance principles for normalized multiple iterated sums and integrals of the form $\mathbb{S}^{(\nu)}(t)=N^{-\nu/2}\sum_{0\leq k_1<...<k_\nu\leq Nt}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$, $t\in[0,T]$ and $\mathbb{S}_N^{(\nu)}(t)=N^{-\nu/2}\int_{0\leq s_1\leq...\leq s_\nu\leq Nt}\xi(s_1)\otimes\cdots\otimes\xi(s_\nu)ds_1\cdots ds_\nu$, where $\{\xi(k)\}_{-\infty<k<\infty}$ and $\{\xi(s)\}_{-\infty<s<\infty}$ are centered stationary vector processes with some weak dependence properties. We show, in particular, that (in both cases) the distribution of $\mathbb{S}^{(\nu)}_N$ is $O(N^{-\delta})$-close, $\delta>0$ in the Prokhorov and the Wasserstein metrics to the distribution of certain stochastic processes $\mathbb{W}_N^{(\nu)}$ 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 $\mathbb{S}_N^{(\nu)}-\mathbb{W}_N^{(\nu)}$ for $\nu=1,2$ and extending these estimates to $\nu>2$ 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.