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Almost sure approximations and laws of iterated logarithm for signatures

Yuri Kifer

TL;DR

This work delivers a direct, general approach to almost sure approximations for normalized multiple iterated sums and integrals (signatures) of weakly dependent processes, establishing strong invariance principles with explicit $p$-variation rates. It treats discrete-time, direct continuous-time, and suspension constructions, and proves laws of the iterated logarithm as well as an almost sure central limit theorem for the entire family of iterated signatures. Central to the methodology are sharp auxiliary estimates, a two-step strong approximation to a Gaussian driver with covariance $\varsigma$, and Chen-type relations that extend the approximation to arbitrarily high iterates. These results generalize previous rough-path–based approaches and make the theory accessible to a broader probabilistic audience, with applications to dynamical systems via suspension flows. The findings provide robust asymptotic descriptions for signatures under weak dependence, with potential impact on statistical analysis of path- and time-series data in dynamical contexts.

Abstract

We obtain strong invariance principles for normalized multiple iterated sums and integrals of the form $\bbS_N^{(ν)}(t)=N^{-ν/2}\sum_{0\leq k_1<...<k_ν\leq Nt}ξ(k_1)\otimes\cdots\otimesξ(k_ν)$, $t\in[0,T]$ and $\bbS_N^{(ν)}(t)=N^{-ν/2}\int_{0\leq s_1\leq...\leq s_ν\leq Nt}ξ(s_1)\otimes\cdots\otimesξ(s_ν)ds_1\cdots ds_ν$, where $\{ξ(k)\}_{-\infty<k<\infty}$ and $\{ξ(s)\}_{-\infty<s<\infty}$ are centered 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 \cite{FK} 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 our paper "Limit theorems for signatures" and we consider a similar setup and rely on many result from there.

Almost sure approximations and laws of iterated logarithm for signatures

TL;DR

This work delivers a direct, general approach to almost sure approximations for normalized multiple iterated sums and integrals (signatures) of weakly dependent processes, establishing strong invariance principles with explicit -variation rates. It treats discrete-time, direct continuous-time, and suspension constructions, and proves laws of the iterated logarithm as well as an almost sure central limit theorem for the entire family of iterated signatures. Central to the methodology are sharp auxiliary estimates, a two-step strong approximation to a Gaussian driver with covariance , and Chen-type relations that extend the approximation to arbitrarily high iterates. These results generalize previous rough-path–based approaches and make the theory accessible to a broader probabilistic audience, with applications to dynamical systems via suspension flows. The findings provide robust asymptotic descriptions for signatures under weak dependence, with potential impact on statistical analysis of path- and time-series data in dynamical contexts.

Abstract

We obtain strong invariance principles for normalized multiple iterated sums and integrals of the form , and , where and 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 \cite{FK} 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 our paper "Limit theorems for signatures" and we consider a similar setup and rely on many result from there.
Paper Structure (23 sections, 16 theorems, 265 equations)

This paper contains 23 sections, 16 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Discrete time case
  4. Straightforward continuous time setup
  5. Continuous time suspension setup
  6. Law of iterated logarithm
  7. An almost sure central limit theorem
  8. Auxiliary estimates
  9. General estimates
  10. The matrix ${\varsigma}$, characteristic functions and partition into blocks
  11. Strong approximations
  12. Discrete time case estimates
  13. Variational norm estimates for sums
  14. Supremum norm estimates for iterated sums
  15. Variational norm estimates for iterated sums
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let (2.4) holds true with integers $L\geq 1$ and a large enough $M$. Then the stationary sequence of random vectors $\xi(n),\,-\infty<n<\infty$ can be redefined preserving its joint distribution on a sufficiently rich probability space which contains also a $d$-dimensional Brownian motion ${\mathcal where the constants ${\varepsilon}_\nu>0$ do not depend on $N$ and $\nu(M)$ depends on $M$ but not

Theorems & Definitions (24)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • Lemma 3.1
  • ...and 14 more