Almost sure approximations and laws of iterated logarithm for signatures

TL;DR

This paper establishes strong invariance principles, laws of iterated logarithm, and almost sure central limit theorems for signatures of stationary processes, extending previous results with more direct methods suitable for dynamical systems.

Contribution

It provides a more accessible, direct approach to invariance principles and limit theorems for signatures, applicable under weak dependence and in continuous time, improving on prior rough paths methods.

Findings

Strong invariance principles for signatures of stationary processes

Laws of iterated logarithm established for these signatures

Almost sure central limit theorem demonstrated for the objects

Abstract

We obtain strong invariance principles for normalized multiple iterated sums and integrals of the form $\bbS_N^{(\nu)}(t)=N^{-\nu/2}\sum_{0\leq k_1<...<k_\nu\leq Nt}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$, $t\in[0,T]$ and $\bbS_N^{(\nu)}(t)=N^{-\nu/2}\int_{0\leq s_1\leq...\leq s_\nu\leq Nt}\xi(s_1)\otimes\cdots\otimes\xi(s_\nu)ds_1\cdots ds_\nu$, where $\{\xi(k)\}_{-\infty<k<\infty}$ and $\{\xi(s)\}_{-\infty<s<\infty}$ are centered stationary vector processes with some weak dependence properties. These imply also laws of iterated logarithm and an almost sure central limit theorem for such objects. 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. Similar results under substantially more restricted conditions were obtained in \cite{FK} relying heavily on rough paths theory and notations while here we obtain these results in a more direct way which makes them accessible to a wider readership. This is a companion paper of our paper "Limit theorems for signatures" and we consider a similar setup and rely on many result from there.