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).
The Central Limit Theorem (CLT) is at the heart of probability and statistics. We can state it in the following way: for a sequence of centred i.i.d. random variables Z = (Z i ) i∈Z with variance σ 2 Z < ∞, there exists an i.i.d. sequence of standard normal Gaussians X = (X i ) i∈Z and a real-valued function 1 f such that
1 For example, one can take f = F -1 Z • Φ where F -1 Z is the quantile function of Z and Φ the standard normal cumulative distribution function.
In many applications of interest, however, the data exhibits non-trivial correlations (see, for example, [Ber92] for a variety of examples) and thus violates the crucial independence assumption. In this context, the celebrated Breuer-Major theorem [BM83] provides a sufficient criterion on the function f and the decay of correlations under which one can still observe CLT behaviour: Theorem 1.1 (Breuer-Major) Consider a stationary sequence X = (X i ) i∈Z of centred, one-dimensional Gaussian random variables with covariance function ρ(i) = E [X 0 X i ] such that ρ(0) = 1. Further, let γ := N (0, 1) and f ∈ L 2 (γ) with Hermite rank d ≥ 1, i.e. its chaos decomposition is given by
where H q denotes the q-th Hermite polynomial. Then, if ρ ∈ ℓ d (Z) and S N is given by
we have
where B is a standard Brownian motion and the variance σ 2 is given by
The previous theorem immediately begs the question as to whether the convergence in (1.1) can be updated to functional convergence in the Skorokhod space D(0, 1). As Chambers and Slud [CS89] have shown by an explicit counterexample, this is not the case under the sole assumptions that f has Hermite rank d and ρ ∈ ℓ d (Z); instead, they provide a sufficient condition, later slightly improved by Ben Hariz [BH02], which requires explicit information on the decay of the Hermite coefficients (c q ) q≥d . Their fast chaos decay assumption, however, is difficult to verify in practice and has been replaced by Nourdin and Nualart [NN20] (see also their joint work with Campese [CNN20]) by the less restrictive, more easily checkable assumption that f ∈ L p 1 (γ) for some p 1 > 2.
Main result. The purpose of the present work is to lift the functional Breuer-Major theorem to the space of rough paths which, in order to be non-trivial, requires to look at R m -valued functions ⃗ f for m ≥ 2. More precisely, we consider the vector of functions
where each f k ∈ L2 (γ) is of the same finite Hermite rank d ≥ 1 with corresponding Hermite decomposition
q H q (x), k ∈ 1, m := {1, . . . , m} .
(1.3)
In addition, from here onwards, we let
i , . . . , X (m) i
)) i∈Z (1.4) be an R m -valued stationary centred Gaussian sequence, that is, a multivariate Gaussian process indexed by Z. In addition, for k, ℓ ∈ 1, m , we let 2
and assume that ρ k,k (0) = 1. Note that ρ k,ℓ (u) = ρ ℓ,k (-u) for u ∈ Z. For t ∈ [0, 1], we then define S N (t) = (S N (t), S N (t)) where the first and second-order processes S N and S N are, respectively, given by
)) .
(1.7)
We refer to Remark 1.4(v) below for further comments on our setting. Finally, we write D k,p (γ) for the Malliavin-Sobolev space w.r.t. γ = N (0, 1) (see Definition 2.6 and Remark 2.13 below) as well as D r-var ([0, 1]; R m ) for the space of Skorokhod-type r-variation rough paths (see Definition 3.3 below).
The following is our main result:
Theorem 1.2 (Rough Functional Breuer-Major Theorem) Let m ∈ N and consider an m-dimensional vector ⃗ f = (f 1 , . . . , f m ) whose components f k ∈ L 2 (γ) each are of Hermite rank at least3 d ≥ 1 with Hermite expansion as in (1.3). We also impose the following conditions:
(1) For any k ∈ 1, m , we have4 f k ∈ D d,2p (γ) for p = 2, i.e. f k ∈ D d,4 (γ).
(2) For each k, ℓ ∈ 1, m , we have i∈Z |ρ k,ℓ (i)| d < ∞, i.e. ρ k,ℓ ∈ ℓ d (Z).
Then, for any r > 2 we have lim
where B = (B, B) is a Brownian rough path with characteristics (Σ, Γ), that is:
(i) The first component B is an m-dimensional Brownian motion with covariance matrix Σ given by Σ = D + 2 Sym(Γ) (1.9) where ∆(u) := E ⃗ f (X 1 ) ⊗ ⃗ f (X u+1 ) , i.e. ∆ k,ℓ (u) = q≥d q! c (k) q c (ℓ) q ρ k,ℓ (u) q (1.10) and D as well as Γ are given by
(1.11)
The second component B is of the form
where the integration denotes Itô integration, cf. Remark 1.4, Point (iv) below.
We record the following corollary which is an immediate consequence of the continuity of the Itô-Lyons map w.r.t. rough paths metric in D r-var ([0, 1]; R m ), see [CFK + 22, Coro. 4.4]. Recall that ⃗ f (X i ) has been introduced in (1.7).
Corollary 1.3 Let n, N ≥ 1 and r > 2. For b ∈ C 3 b (R n ; R n ) and V ∈ C 3 b (R n ; R n×m ), let Y N be defined through the Euler-type recurrence relation
whenever i ∈ 0, N -1 , which is extended to t ∈ [0, 1] by setting Y N (t) := Y N i N whenever i/N ≤ t < (i + 1)/N . For Γ ∈ R m×m given in (1.11), define c :
If y N → y as N → ∞, then Y N converges in law in r-variation topology to the unique strong solution of the SDE
where B is an R m -valued Brownian motion with covariance
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