Large Deviations for Iterated Sums and Integrals

Abstract

We describe large deviations for normalized multiple iterated sums and integrals of the form $\bbS_N^{(ν)}(t)=N^{-ν}\sum_{0\leq k_1<...<k_ν\leq Nt}ξ(k_1)\otimes\cdots\otimesξ(k_ν)$, $t\in [0,T]$ and $\bbS_N^{(ν)}(t)=N^{-ν}\int_{0\leq s_1\leq...\leq s_ν\leq Nt}ξ(s_1)\otimes\cdots\otimesξ(s_ν)ds_1\cdots ds_ν$, where $\{\hat{I}¾(k){\}}_{-\infty <k<\infty }$ and $\{\hat{I}¾(s){\}}_{-\infty <s<\infty }$ are centered bounded stationary vector processes whose sums or integrals satisfy a trajectorial large deviations principle.

5 pages