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Large Deviations for Iterated Sums and Integrals

Yuri Kifer, Ofer Zeitouni

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 $\{ξ(k)\}_{-\infty<k<\infty}$ and $\{ξ(s)\}_{-\infty<s<\infty}$ are centered bounded stationary vector processes whose sums or integrals satisfy a trajectorial large deviations principle.

Large Deviations for Iterated Sums and Integrals

Abstract

We describe large deviations for normalized multiple iterated sums and integrals of the form , and , where and are centered bounded stationary vector processes whose sums or integrals satisfy a trajectorial large deviations principle.

Paper Structure

This paper contains 4 sections, 2 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs
  4. Law of large numbers

Key Result

Theorem 2.2

Let the Assumption as hold true. Then $S_n^{(\nu)}$ satisfies LDP in $C([0,T];\,{\mathbb R}^{d\nu})$ with the good rate function $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (3)

  • Theorem 2.2
  • Proposition 2.3
  • proof