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A Rough Functional Breuer-Major Theorem

Henri Elad Altman, Tom Klose, Nicolas Perkowski

Abstract

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with Meyer's inequality and a Kolmogorov-type criterion for the r-variation of cadlag rough paths, due to Chevyrev et al. (2022). Since martingale techniques do not apply, we obtain the convergence of the finite-dimensional distributions through a bespoke version of Slutsky's lemma: First, we overcome the lack of hypercontractivity by an iterated integration-by-parts scheme which reduces the remaining analysis to finite Wiener chaos; crucially, this argument relies on Malliavin differentiability of the nonlinearity but not on chaos decay and, as a consequence, encompasses the centred absolute value function. Second, in the spirit of the law of large numbers, we show that the diagonal of the second-order process converges to an explicit symmetric correction term. Finally, we compute all the moments of the remaining process and, through a fine combinatorial analysis, show that they converge to those of the Stratonovich Brownian rough path perturbed by an antisymmetric area correction, as computed by a suitable amendment of Fawcett's theorem. All of these steps benefit from a major combinatorial reduction that is implied by the original argument of Breuer and Major (1983).

A Rough Functional Breuer-Major Theorem

Abstract

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with Meyer's inequality and a Kolmogorov-type criterion for the r-variation of cadlag rough paths, due to Chevyrev et al. (2022). Since martingale techniques do not apply, we obtain the convergence of the finite-dimensional distributions through a bespoke version of Slutsky's lemma: First, we overcome the lack of hypercontractivity by an iterated integration-by-parts scheme which reduces the remaining analysis to finite Wiener chaos; crucially, this argument relies on Malliavin differentiability of the nonlinearity but not on chaos decay and, as a consequence, encompasses the centred absolute value function. Second, in the spirit of the law of large numbers, we show that the diagonal of the second-order process converges to an explicit symmetric correction term. Finally, we compute all the moments of the remaining process and, through a fine combinatorial analysis, show that they converge to those of the Stratonovich Brownian rough path perturbed by an antisymmetric area correction, as computed by a suitable amendment of Fawcett's theorem. All of these steps benefit from a major combinatorial reduction that is implied by the original argument of Breuer and Major (1983).
Paper Structure (40 sections, 52 theorems, 347 equations, 4 figures)

This paper contains 40 sections, 52 theorems, 347 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main result.
  3. Discussion and relation to the literature.
  4. Strategy of proof, key novelties, and insights.
  5. Organisation of the article.
  6. Acknowledgements.
  7. Preliminaries and auxiliary results
  8. Notation
  9. Malliavin calculus
  10. The Hermite shift operator
  11. Diagram formula, regular, and irregular diagrams
  12. Convergence of normalised sums of covariances
  13. Tightness
  14. Variation topologies on rough paths
  15. Tightness result
  16. ...and 25 more sections

Key Result

Theorem 1.1

Consider a stationary sequence $X = (X_i)_{i \in \mathbb{Z}}$ of centred, one-dimensional Gaussian random variables with covariance function $\rho(i) = \mathbb{E}\sbr[0]{X_0 X_i}$ such that $\rho(0) = 1$. Further, let $\gamma \coloneqq \mathcal{N}(0,1)$ and $f \in L^2(\gamma)$ with Hermite rank $d \ where $H_q$ denotes the $q$-th Hermite polynomial. Then, if $\rho \in \ell^d(\mathbb{Z})$ and $S_N$

Figures (4)

  • Figure 1: Visualisation of diagrams. Note that, by relabelling the vertices within levels, a regular diagram $G$ can always be represented as in Subfigure \ref{['subfig:regular_diagram']} because this does not change $\mathfrak{C}_G$ as introduced in \ref{['e:diagram_formula:CG']} below.
  • Figure 2: Two instances of the worst-case scenario in terms of actual required derivatives, namely $3d$. In the right figure, however, the difference is that the the commutation relation between $D$ and $\delta$, Lemma \ref{['lem:commutation_D_delta']}, requires $4d$ derivatives in an intermediate step to deal with $D^{3d} \delta^{d} \del[0]{\sbr[0]{\mathcal{S}_{d}h_4}(W(e_{\boldsymbol{\underline{\iota}}_4}))e_{\boldsymbol{\underline{\iota}}_4}^{\otimes d}}$.
  • Figure 3: Example of the conversion mechanism into Feynman diagrams.
  • Figure 4: Structure of regular diagrams.

Theorems & Definitions (137)

  • Theorem 1.1: Breuer--Major
  • Theorem 1.2: Rough Functional Breuer--Major Theorem
  • Corollary 1.3
  • Remark 1.4: General comments
  • Remark 1.5: Differentiability, integrability, and summability assumptions
  • Definition 2.1: Isonormal Gaussian process
  • Lemma 2.2
  • Definition 2.3: Hermite polynomials and Wiener chaos
  • Definition 2.4: Wiener--Itô isometry
  • Definition 2.5: Malliavin derivative
  • ...and 127 more