Almost sure approximations and laws of iterated logarithm for signatures

Abstract

We obtain strong invariance principles for normalized multiple iterated sums and integrals of the form $S_N^{\left(\nu \right)}\left(t\right)=N^{-\nu /2}{\sum }_{0\le k_1<...<k_{\nu }\le Nt}\xi \left(k_1\right)\otimes \cdots \otimes \xi \left(k_{\nu }\right)$, $t\in [0,T]$ and $S_N^{\left(\nu \right)}\left(t\right)=N^{-\nu /2}{\int }_{0\le s_1\le \cdots \le s_{\nu }\le Nt}\xi \left(s_1\right)\otimes \cdots \otimes \xi \left(s_{\nu }\right)d{s}_1\cdots d{s}_{\nu }$, where ${\left\{\xi \left(k\right)\right\}}_{-\infty <k<\infty }$ and ${\left\{\xi \left(s\right)\right\}}_{-\infty <s<\infty }$ 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 Friz and Kifer (2024) 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 Kifer (0000) and we consider a similar setup and rely on many result from there.