Large Deviations for Iterated Sums and Integrals

We describe large deviations for normalized multiple iterated sums and integrals of the form ${\text{bbS}}_N^{(\nu )}(t)=N^{-\nu }{\sum }_{0\le k_1<\mathrm{.}\mathrm{.}\mathrm{.}<k_{\nu }\le Nt}\xi (k_1)\otimes \cdots \otimes \xi (k_{\nu })$, $t\in [0,T]$ and ${\text{bbS}}_N^{(\nu )}(t)=N^{-\nu }{\int }_{0\le s_1\le \mathrm{.}\mathrm{.}\mathrm{.}\le s_{\nu }\le Nt}\xi (s_1)\otimes \cdots \otimes \xi (s_{\nu })d{s}_1\cdots d{s}_{\nu }$, where $\{\xi (k){\}}_{-\mathrm{\infty }<k<\mathrm{\infty }}$ and $\{\xi (s){\}}_{-\mathrm{\infty }<s<\mathrm{\infty }}$ are centered bounded stationary vector processes whose sums or integrals satisfy a trajectorial large deviations principle.