Large Deviations for Iterated Sums and Integrals

TL;DR

This paper establishes large deviation principles for normalized iterated sums and integrals of stationary vector processes, extending the understanding of their probabilistic behavior in the asymptotic regime.

Contribution

It provides a rigorous large deviations framework for multiple iterated sums and integrals of stationary processes, which was previously not well-understood.

Findings

Large deviations principles are derived for iterated sums and integrals.

Results apply to centered bounded stationary vector processes.

The framework extends large deviations theory to complex iterated structures.

Abstract

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