Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields

Let $B=(B_x)_{x\in\mathbb{R}^d}$ be a collection of $N(0,1)$ random variables forming a real-valued continuous stationary Gaussian field on $\mathbb{R}^d$, and set $C(x-y)=\mathbb{E}[B_xB_y]$. Let $\varphi:\mathbb{R}\to\mathbb{R}$ be such that $\mathbb{E}[\varphi(N)^2]<\infty$ with $N\sim N(0,1)$, let $R$ be the Hermite rank of $\varphi$, and consider $Y_t = \int_{tD} \varphi(B_x)dx$, $t>0$, with $D\subset \mathbb{R}^d$ compact. Since the pioneering works from the 80s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for $Y_t$ have been constantly refined, extended and applied to an increasing number of diverse situations, to such an extent that it has become a field of research in its own right. The common belief, representing the intuition that specialists in the subject have developed over the last four decades, is that as $t\to\infty$ the fluctuations of $Y_t$ around its mean are, in general (i.e. except possibly in very special cases), Gaussian when $B$ has short memory, and non Gaussian when $B$ has long memory and $R\geq 2$. We show in this paper that this intuition forged over the last forty years can be wrong, and not only marginally or in critical cases. We will indeed bring to light a variety of situations where $ Y_t $ admits Gaussian fluctuations in a long memory context. To achieve this goal, we state and prove a spectral central limit theorem, which extends the conclusion of the celebrated Breuer-Major theorem to situations where $C\not\in L^R(\mathbb{R}^d)$. Our main mathematical tools are the Malliavin-Stein method and Fourier analysis techniques.


Introduction
Fix a dimension d ≥ 2, and consider a real-valued almost surely continuous Gaussian field (B x ) x∈R d defined on R d . Assume furthermore that B is stationary, that is, there is a function C : R d → R such that (1) Cov(B x , B y ) = C(x − y), x, y ∈ R d , and suppose that B x ∼ N(0, 1) for all x ∈ R d or, equivalently, that E[B x ] = 0 and C(0) = 1. As a second ingredient, consider a measurable function ϕ : R → R such that (2) E ϕ(N) 2 < ∞, for N ∼ N(0, 1).
Our object of interest in this paper is where D ⊂ R d is compact with Vol(D) > 0 and tD := {tx|x ∈ D}.
Well-posedness of (3) as a random variable in L 2 (Ω) is ensured by Proposition 3 and the almost sure continuity of (B x ) x∈R d . We also observe that the continuity of B, together with its stationarity, implies 1 the continuity of its covariance function C, a property that will be needed to evoke Bochner's theorem later in (14).
Studying the asymptotic behavior of functionals of the form (3) dates back from the eighties, with seminal works by Breuer and Major [4], Dobrushin and Major [6], Rosenblatt [25] and Taqqu [27]. Since then, limit theorems for (3) have been constantly investigated, and represent nowadays a central theme in the modern probability theory.
We note that many interesting geometric quantities associated with the Gaussian field (B x ) x∈R d can be represented as functionals of the form (3). For instance, the choice ϕ = 1 (−∞,u] (resp. ϕ = 1 [u,∞) ), u ∈ R corresponds to the volume of the lower (resp. upper) level sets of (B x ) x∈R d .
Since (2) holds, we can decompose ϕ in Hermite polynomials (see, e.g., [17,Section 1.4]) as (4) ϕ = E[ϕ(N)] + ∞ q=R a q H q , with R ≥ 1 such that a R = 0, where H q denotes the qth Hermite polynomial and a q = a q (ϕ) = 1 q! E [ϕ(N)H q (N)] ∈ R. The integer R ≥ 1 is called the Hermite rank of ϕ. We also define the second Hermite rank R ′ ≥ 2 of ϕ as the Hermite rank of ϕ( In the present paper, we are more specifically interested in the asymptotic behavior of where we have Y t ∈ L 2 (Ω) for all t, and where we note m t = E[Y t ] and σ t = Var(Y t ) > 0. For simplicity, to ensure that σ t > 0 for all t, we will assume for the rest of the paper the existence of some k ≥ 1 such that a 2k = 0 2 , or equivalently that ϕ is not an odd function. As an illustration of what may happen when ϕ is odd, see (22).
Throughout all the paper, for two functions f, g : R + → R + we write to indicate that f (t) = O(g(t)) and g(t) = O(f (t)) as t → ∞.
1.1. Previous results. Given its importance in our paper, we start with the celebrated Breuer-Major theorem, stated here in its continuous form.
Theorem 1 (Breuer,Major [4]). Let B = (B x ) x∈R d be a real-valued continuous centered Gaussian field on R d , assumed to be stationary and to have unit-variance. Let ϕ : R → R be such that E[ϕ(N) 2 ] < ∞ with N ∼ N(0, 1), let R be the Hermite rank of ϕ, consider Y t defined by (3), and recall the definition (1) of the covariance function C.
To describe the asymptotic behavior in the case where C ∈ L R (R d ) (that is, when we cannot apply the Breuer-Major Theorem 1), we have to be more precise on the behavior of C at infinity. In the papers studying limit theorem for (5) in a general framework (i.e. not for a particular model), it is often (if not always) assumed that (8) C(x) = |x| −β L(|x|), with β ∈ (0, ∞) and L : R + → R slowly varying (that is, satisfying L(λr)/L(λ) → 1 as λ → ∞, for every fixed r > 0). The following three different situations (i)-(ii)-(iii) then occur: (i) If β > d R then C ∈ L R (R d ) and we say that we are in the shortmemory case. We deduce from Breuer-Major Theorem 1 that law → N(0, σ 2 ). (ii) If β = d R we are in a critical case. Although C ∈ L R (R d ), fluctuations of Y t around its mean are still asymptotically Gaussian (see, e.g., [4,Theorem 1'], [15,Section 5] and [16,Theorem 1]). (iii) If β < d R then C ∈ L R (R d ) and we say that we are in the long-memory case. A theorem of Dobrushin and Major [6] asserts that t −(d− Rβ law → Z where, up to a multiplicative constant, Z is the Hermite distribution of order R and self-similarity index H ∈ (0, 1) (with H depending only on β). Since we do not use it in the sequel, we do not give its precise definition here. (The interested reader can e.g. consult [28].) Let us only stress here that the Hermite distribution of order R belongs to the Rth Wiener chaos, and so is not Gaussian as soon as R ≥ 2.

1.2.
Motivating examples. The intuition we can naturally develop from the previous (i)-(ii)-(iii) (and that represents the common intuition forged by the papers written on the subject over the last forty years) is that Y t defined by (3) displays Gaussian (resp. non-Gaussian) fluctuations when the Gaussian field B has short (resp. long) memory, and this whatever the function ϕ with Hermite rank R ≥ 2. (The case R = 1 is apart. 3 ) As anticipated, we will show in this paper that this intuition can be wrong, and not only marginally or in critical cases. We will indeed state a central limit theorem (Theorem 2 below) whose conclusion is valid provided that a certain spectral condition is satisfied. As we will see, this may lead to Gaussian fluctuations in a long memory context, in total contrast with Dobrushin and Major [6] (see (iii) above).
Recently, Berry's Random Wave Model (BRWM) has attracted a lot of attention. It is defined as follows. Choose the dimension d = 2 and consider the centered continuous Gaussian field B on R 2 with covariance E[B x B y ] = C(x − y) = J 0 (|x − y|), with J 0 the Bessel function of the first kind of order 0 (see Section 2.2 for its definition and some properties); in particular, we have as |x| → ∞, see e.g. [11,Theorem 4]. This field, called in this way in honor of Berry who introduced it in the seminal paper [2], can be seen as a universal Gaussian field emerging as the local scaling limit of a number of random fields on two-dimensional manifolds, see e.g. [5] and the references therein. It is widely regarded as a popular model for the Laplacian eigenfunctions (with large eigenvalue t 2 ) of classically chaotic billiards, hence its importance in quantum mechanics. Indeed, integrating ϕ(B x ) over tD is the same as integrating ϕ(B tx ) over the fixed domain D, after a change of variable. As we will see, our Theorem 2 will allow to prove the Gaussian fluctuations of D ϕ(B tx )dx for many functionals that were never investigated in the literature, in particular for the cases R = 2 and R = 1 in (10), enriching the knowledge on the geometric properties of the Berry's random wave model. Indeed, to the best of our knowledge, so far Gaussian fluctuations were proved only for (3) under the assumption R ≥ 4 (see e.g. [21]), or for nodal set volumes (see [19]), where the latter cannot be expressed in the form (3).
If we compare (9) with (8), it is like we have β = 1 2 and that we had replaced the slowly varying function L in (8) by the bounded and oscillatory function cos(· − π 4 ). As we will see, this replacement is all but insignificant in the presence of long memory. Indeed, taking for example D compact such that D =D and with smooth ∂D and nonvanishing Gaussian curvature, if we apply our main result (Theorem 2 below) for R ∈ {1, 2, 4} and Breuer-Major theorem for R ≥ 5, we obtain (recalling that R, resp. R ′ , denotes the Hermite rank, resp. the second Hermite rank, of the non-odd function ϕ): if R ≥ 5 or (R = 1 and R ′ ≥ 5) and (11) Indeed, Gaussian fluctuations (11) and asymptotics (10) follow directly from (9) and the Breuer-Major Theorem 1 when R ≥ 5, because in this case we have R d |C(x)| R dx < ∞. The case R = 4 is comparable with the situation (ii) in the model (8), since β = 1 2 = d R ; Gaussian fluctuations (11) and asymptotic (10) displaying a logarithmic correction are then not surprising, since in agreement with what is usually observed in critical cases. In constrast, the fact that we still have Gaussian fluctuations in (11) with unconventional rate when R ∈ {1, 2} is very surprising. Indeed, since β = 1 2 < 1 ≤ d R in this case, we are in the longmemory case where non-Gaussian fluctuations are usually the rule, see (iii). This last case turns out to be the most important, since R = 1 and R = 2 are the most common values in applications.
As we will see, what we just described (Gaussian fluctuations in a long memory framework) for BRWM is not an isolated phenomenon. In fact, we are going to state and prove Theorem 2 (our main result), which not only explains in a clear way the phenomenon observed for BRWM, but also bring to light an easy-to-check condition on the spectral measure of a general Gaussian field B (see (20) below) to imply Gaussian fluctuations for Y t .
We conclude this section by emphasing that Gaussian fluctuations in presence of long memory have already been observed in the literature in other (non-Euclidean) contexts, in particular for integral functionals of random Laplace eigenfunctions on the sphere S 2 , see [14] for more details.
1.3. Main result. In order to state Theorem 2, we have to introduce a certain number of further quantities and notations. We continue to let B, ϕ and Y t be as described in (1), (2) and (3) respectively, and we recall the Hermite decomposition (4) of ϕ defining its Hermite rank R and its second Hermite rank R ′ .
Fix t > 0 and q ≥ R, and set Using (4) , we immediately get that Moreover, a direct computation (making use of the isometry properties of Hermite polynomials, see, e.g., [17,Section 1.4]) yields that Since our field B is stationary and its covariance function C is continuous, Bochner's theorem yields the existence of a finite measure G on R d such that (14) C with ·, · the usual scalar product on R d . The measure G is called the spectral measure of B, and it will be our gateway towards the Fourier analysis techniques developed in the sequel. At this stage, let us make a further assumption on B, by supposing that it is also isotropic. In our framework, this is equivalent to suppose that the quantity C(x) only depends on the norm |x|, namely that there is a function ρ : R + → R such that Since µ is increasing and bounded, it defines a finite measure on R + , called the isotropic spectral measure of B. Because C(x) only depends of |x|, we can write, with w d = S d−1 dξ (S d−1 being the unit sphere of R d ): where (17) b d (|λ|) = 1 w d S d−1 e i λ,ξ dξ.
By d-dimensional polar coordinates, we readily find (see [26, page 815]) that , r > 0, with J ν denoting the Bessel function of the first kind of order ν (see Section 2.2) and c d > 0 a constant depending only of d.
We are at last in a position to state our main result, which we have decided to call the spectral central limit theorem, because it leads to Gaussian fluctuations on the one hand (hence 'central') and the main assumption we have to check is the spectral condition (20) on the other hand (hence 'spectral').
Var(Y t ) > 0, and recall the definition (15) of the isotropic spectral measure µ. Set (19) w q,t = {|z|≤t} C(z) q dz, and assume that the following spectral condition holds Finally, when R = 2 assume that |F as |x| → ∞, as |x| → ∞, with F the Fourier transform (since we assumed D to be compact, these two assumptions are satisfied for example when D =D and ∂D is smooth with non-vanishing Gaussian curvature, see e.g. [3]). Then we have: By a simple Fubini argument, we observe that the spectral condition (20) is equivalent to When ρ(r) = r −β L(r) with β ∈ (0, d R ) and L slowly varying as r → ∞, we have µ(s) ∼ s β L(s −1 ) as s → 0, see [12,Thm 1.4.3]. We deduce that (20)- (21) is not satisfied, which is of course perfectly consistent with the conclusion of the Dobrushin-Major noncentral limit theorem [6].
Also, let us observe that the assumption "ϕ is not odd" is equivalent to "∃ k ≥ 1 such that a 2k = 0", and so is needed only in the case where R = 1 and R ′ ≥ 5. Moreover, the cases not covered by our Theorem 2 are R ≥ 3 odd, (R, R ′ ) = (1, 3) and ϕ odd. In these situations, very peculiar things can happen. Consider for example ϕ(x) = x, which gives Y t = tD B x dx. Using (29), (14) and Fubini we have, with F the Fourier transform, Using that g tD = 1 tD * 1 −tD we obtain (22) Var In particular, for D = {|z| ≤ 1} ⊂ R d and Berry's Random Wave Model B = (B x ) x∈R 2 we get (using (26)) ; although Y t is Gaussian the conclusion of Theorem 2 cannot hold in this case, the limit of Yt−mt σt being ill-defined due to the fact that Card{t : To understand the significance of our spectral CLT, let us go back to BRWM and explain how Theorem 2 together with Breuer-Major theorem allow to prove (11). Since C(x) = ρ(|x|) = J 0 (|x|), it follows immediately from the representation (16) that the isotropic spectral measure associated with BRWM B = (B x ) x∈R 2 is µ = δ 1 , with δ 1 the Dirac mass at 1. In particular, the spectral condition (20) is obviously satisfied, whatever the value of R. The convergence (11) thus follows from the Breuer-Major Theorem 1 (see also (9)) if R ≥ 5 and from Theorem 2 in all the other cases in (10).

1.4.
Possible natural extensions of Theorem 2. As a natural extension of the present work, it would be interesting to study the joint convergence associated with Theorem 2. More precisely, taking D 1 , . . . , D n compact domains in R d , is it possible to identify conditions that resemble those in Theorem 2 ensuring that the random vector converges, after proper normalization, to a Gaussian random vector? In other words, can we prove a multivariate spectral central limit theorem? Such an extension is not immediate, because the spectral condition (20) alone looks too general to capture the asymptotic behavior of the covariances between the components of the random vector. A partial answer to this question (covering also Berry's model) will be given in the forthcoming work [13] by the first author. Note that a multivariate result was obtained in the particular case of the nodal length of Berry's model restricted to a finite collection of smooth compact domains D 1 , . . . , D n of the plane in the recent paper [24].
A further stronger generalization of Theorem 2 would be to prove a functional spectral central limit theorem. This problem has been investigated in the particular case of nodal set volumes in [20] by ad hoc techniques. In our general framework, we believe however that proving such a functional extension would require novel ideas that go beyond the scope of the present paper.
1.5. Plan of the paper. Apart from Section 6 that illustrates a use of Theorem 2, the rest of the paper is fully devoted to the proof of this latter. More precisely, after a few needed preliminaries given in Section 2, (i) in Section 3 we will first consider the situation where the Hermite rank R of ϕ is even and bigger or equal than 4. As we will see, in this case we have that tD |C(x)| R dx = O(log t), meaning that we are somehow in the 'domain of attraction' of the Breuer-Major theorem 1.; (ii) we will then deal with the remaining cases, namely R = 2 in Section 4 and (R = 1, ϕ non-odd and R ′ = 3) in Section 5. We regard this part as the most important contribution of this paper, especially since among the Hermite ranks, R = 1 and R = 2 are the most common values encountered in practice. They turn out to be also the most difficult cases. To deal with them, we will have to introduce novel ideas, by making heavy use of Fourier techniques in a way that, to the best of our knowledge, has never been introduced before this work.

2.
A few preliminaries for the proof of Theorem 2 The following proposition explains why the random variable Y t defined by (3) is well-defined for all t.
Consider the Hermite expansion of ϕ given by (4), and set For each fixed t > 0, the sequence tD ϕ n (B x )dx, n ≥ R, is a.s. welldefined and converges in L 2 (Ω). The limit is noted Y t and we write, possibly with a slight abuse of notation: Proof. That tD ϕ n (B x )dx is a.s. well-defined is because tD is compact, ϕ n is a polynomial and B is continuous. For any n, m ≥ R and q ∈ N, That is, tD ϕ n (B x )dx n≥R is an L 2 (Ω)-Cauchy sequence, and the desired conclusion follows.

2.2.
Bessel functions and Fourier transform of the indicator of the unit ball. The Bessel function J ν , ν ≥ 0, is defined by the series and it is a solution to the ODE It satisfies the classical Schläfli's representation as well as the Mehler-Sonine formula Schläfli's representation (23) immediately implies that J ν is bounded on R. This, combined with an inequality found e.g. in [11] (see also the references therein), leads to (25) sup a simple property that we will use several times in the forthcoming proof of Theorem 2.
The Fourier transform of the indicator of the unit ball is given by the last equality being a consequence of (24).

2.3.
Reduction to the Rth chaos. In the short memory case, that is when C ∈ L R (R d ), the Breuer-Major theorem 1 yields Gaussian fluctuations for Y t . In its modern proof given by [18] (see also [17,Chapter 7] and the proof goes as follows. It is first shown that t −d σ 2 t → σ 2 in (7) by means of the isometry property of Hermite polynomials. Then, it is proved using the Fourth Moment Theorem (see [17,Theorem 5 [17,Theorem 5.2.7]. In particular, we observe that no term is asymptotically dominant in (27): they all contribute to the limit.
As the following result will show, the situation is totally opposite in the critical and long memory cases (when R is even): here, it is the term Y R,t alone which is responsible of the limit.
Proposition 4 (Reduction to the Rth chaos). Let B = (B x ) x∈R d be a real-valued continuous centered Gaussian field on R d , and assume that B is stationary and has unit-variance (note that d ≥ 2 and isotropy are not required here). Let ϕ : R → R be such that E[ϕ(N) 2 ] < ∞ (with N ∼ N(0, 1)) and have Hermite decomposition (4) and Hermite rank R, and consider Y t defined by (3), where D ⊂ R d compact and Var(Y t ) > 0, and recall the definition (1) of C(x) and the definition (12) Applying the change of variable z = y − x and then Fubini, we obtain that Applying Cauchy-Schwarz inequality we obtain, for q > R, Applying Cauchy-Schwarz again, but this time with 2q−R > R instead of q > R, we obtain .
By iterating the process, we get, for every n ≥ 3 When q > R, we have R + 2 n (q − R) ≥ 2 n , so we may and will choose n large enough so that R + 2 n (q − R) ≥ M for all q > R (recall from the statement of Proposition 4 that M is an integer supposed to be such that Combining all these facts together, we finally get that To conclude the proof of Proposition 4, it remains to prove (28).
Var(Y q,t ), we deduce from (31) and (32) that and this tends to zero as t → ∞. By plugging this into (34) and taking into account that (33) holds, we deduce that from which the implication (28) now follows easily.

Elements of Malliavin calculus and the Fourth Moment theorem.
To obtain the N(0, 1) distribution in the limit in Theorem 2, we rely on the Fourth Moment Theorem of Nualart and Peccati (see [17,Theorem 5.2.7]). Before stating and reformulating it in our framework, we start with some notions on Malliavin calculus. For all the missing details we refer to [17] and [22].
2.4.1. The Wiener-Itô integral. Let B = (B x ) x∈R d be a real-valued continuous centered Gaussian field on R d , and assume that B is stationary with unit-variance. Define Since H is a real, separable Hilbert space, there is an isometry Φ : Consider now the Gaussian noise W = {W (h), h ∈ L 2 (R + )}, i.e. a family of centered Gaussian random variables with covariance given by Since in this paper we are only interested in distributions, we can assume without loss of generality that (B x ) x∈R d = (W (e x )) x∈R d .
For q ≥ 1, we also define the qth Wiener chaos as the linear subspace of L 2 (Ω) generated by {H q (W (h)), h ∈ L 2 (R + )}. For every h ∈ L 2 (R + ) with h L 2 (R + ) = 1, we define the qth Wiener-Itô integral Note that the definition of I q can be extended to every function in the space L 2 s (R q + ) of symmetric functions in L 2 (R q + ) so that I q : L 2 s (R q + ) → L 2 (Ω) is a linear map, because span{h ⊗q , h ∈ L 2 (R + )} is dense in L 2 s (R q + ) (see, e.g., [7]).

Contractions and the Fourth Moment Theorem.
For q ∈ N, r ∈ {1, . . . , q−1} and h 1 , h 2 symmetric functions with unit norm in L 2 (R + ), we can define the rth contraction of h ⊗q 1 and h ⊗q 2 as the (non-symmetric) element of L 2 (R 2q−2r + ) given by Here again this definition can be extended (taking the closure in L 2 (R q + )) to every h 1 , h 2 in L 2 s (R q + ). We will denote the norm in the space L 2 (R q + ) as · q . We can finally state the celebrated Fourth Moment Theorem, proved in [23] by Nualart and Peccati.
Theorem 5 (Fourth Moment Theorem). Fix q ≥ 2, consider (h t ) t>0 ⊂ L 2 s (R q + ) and assume that E[I q (h t ) 2 ] → 1 as t → ∞. Then the following assertions are equivalent: • I q (h t ) converges in distribution to a standard Gaussian N ∼ N(0, 1).
An important consequence of the previous result (in our framework) is the following. Theorem 6. Let B = (B x ) x∈R d be a real-valued continuous centered Gaussian field on R d , and assume that B is stationary and has unitvariance. Assume that D ⊂ R d is compact with Vol(D) > 0. Recall the definition (1) of C. Fix also q ≥ 2, recall the definition (12) of Y q,t and assume Var(Y q,t ) > 0 for all t large enough. If we have, for any Proof. Let us first write Y q,t as a q-th multiple Wiener-Itô integral with respect to B: Applying the change of variable x = x 3 − x 1 , y = x 2 − x 4 , z = x 1 − x 2 , a = x 4 (whose Jacobian is equal to 1) and using the symmetry of C, we get that The Fourth Moment Theorem asserts that Y q,t / Var(Y q,t ) converges in distribution to N(0, 1) if (and only if) f t,q ⊗ r f t,q 2q−2r /Var(Y q,t ) → 0 for all r = 1, . . . , q − 1. The desired conclusion thus follows from (35) and the previous bound for f t,q ⊗ r f t,q 2 2q−2r .

Proof of Theorem 2 when R ≥ 4 even
This section is devoted to the proof of Theorem 2 when R ≥ 4 is even, which is equivalent to say that d R ≤ d−1 2 and R even. It represents the 'easy' part of Theorem 2.
To ease the exposition, we write in the following proposition the statement obtained when, in Theorem 2, we suppose that d R ≤ d−1 2 and R even.
If (20) holds, if R even and d R ≤ d−1 2 (i.e. R ≥ 4 and R even), then The goal of this Section 3 is to prove Proposition 7. As we will see, it will be a direct consequence of Lemma 8 and Proposition 9 below.
We start with Lemma 8.

Proposition 9. Let B = (B x ) x∈R d be a real-valued continuous centered
Gaussian field on R d , and assume that B is stationary and has unitvariance (note that we did not assume isotropy and d ≥ 2). Let ϕ : If R is even, if lim t→∞ w R,t = ∞ and if Proof. The proof is divided into several steps. Recall the definition (12) of Y q,t .
Step 1. We claim that if R even, then where v R,t is defined by (29). To prove it, below we let c > 0 denote a constant independent of t whose value can change from one instance to another. Using (29), we have on the one hand where the last equality follows from the so-called doubling conditions at the origin for non-negative positive definite functions (see [10]).
On the other hand, g D is uniformly continuous (in particular continuous in 0) according to [8]. We deduce that g D z t ≥ Vol(D) − 1 2 Vol(D) = 1 2 Vol(D) for all z ∈ {|z| ≤ δ D t} for some δ D > 0 depending only on D and for every t > 0. As a result, where the last inequality follows again from the doubling conditions proved in [10]. The announced claim (38) follows.
Step 2. Since w R,t → ∞, it follows from Step 1 that Applying Proposition 4, we obtain that σ 2 t ≍ t d w R,t and also that (37) will follow if we prove that Y R,t / Var(Y R,t ) → N(0, 1).
Step 3. In this last step, we prove that Y R,t / Var(Y R,t ) → N(0, 1), which will complete the proof of Proposition 9, see the conclusion of Step 2. To do this, we use Theorem 6, following the same approach as in [19,Lemma 8.1]. Fix a contraction index r ∈ {1, . . . , R − 1}. Using the inequality u r v R−r ≤ u R + v R for u, v ∈ R + , we have where the last inequality follows from the change of variable a = x+y+z and doubling conditions in [10], and We deduce that (35) is bounded by where the big O comes from (38). We deduce from (36) that w q,t = O t d−d q R for q < R. Using this last fact, we get and then (35) converges to 0 as t → ∞. Therefore, the convergence Y R,t / Var(Y R,t ) → N(0, 1) follows from Theorem 6, and completes the proof of Proposition 9.
We can now proceed with the proof of Proposition 7, that is, the proof of Theorem 2 when R ≥ 4 and R even (or equivalently, d R ≤ d−1 2 and R even).
Proof of Proposition 7. If w R,t is convergent, then the result follows applying the Breuer-Major theorem 1. So we can assume that w R,t → ∞. Now, by comparing the statements of Proposition 7 and Proposition 9, we see that we are left to check that, if (20) holds with d R ≤ d−1 2 , then (36) holds. Since this is a mere application of Lemma 8 with β = d R , the proof of Proposition 7 is complete.

Proof of Theorem 2 when R = 2
This section is devoted to the proof of Theorem 2 when R = 2. It requires the introduction of novel ideas with respect to the existing literature, mainly Fourier arguments.
To ease the exposition, we write in the following proposition the statement obtained when, in Theorem 2, we additionally suppose that Proposition 10. Fix d ≥ 2, let B = (B x ) x∈R d be a real-valued continuous centered Gaussian field on R d , and assume that B is stationary, isotropic and has unit-variance. Let ϕ : R → R be such that E[ϕ(N) 2 ] < ∞ with N ∼ N(0, 1), and assume that ϕ has Hermite rank R = 2. Consider Y t defined by (3), where D compact and Vol(D) > 0. Set m t = E[Y t ] and σ t = Var(Y t ) > 0, and recall the definition (15) of the isotropic spectral measure µ and the definition (19) of w q,t .
as |x| → ∞ 4 , with F the Fourier transform, and if the spectral condition holds Before proving Proposition 10 (which is the goal of this section), let us recall the definition (12) of Y 2,t and let us state and prove some preliminary results. We start with Lemma 11, reformulating in a spectral form the norm of the contractions introduced in section 2.4. Note that the following result has an analogous version for a general rth contraction in the case Y q,t , but we skip this unnecessary extension for the sake of brevity.
Lemma 11. Let B = (B x ) x∈R d be a real-valued continuous centered Gaussian field on R d , and assume that B is stationary and has unitvariance (note that we did not assume isotropy and d ≥ 2). Assume that D ⊂ R d is compact with Vol(D) > 0, recall the notions introduced in section 2.4 and write x dx.

Recall the definition (1) of C and define
Finally, recall the definition (14) of G. Then where F is the Fourier transform, and Proof. Proceeding exactly as in the proof of Theorem 6, we have Applying the change of variable where in the last equality we used the translation invariance of the Lebesgue measure (subtracting w) and the definition (40), justified by the fact that tD ∩ (tD + a) is empty when |a| > diam(D)t. Now recall the definition (41). We have where in the last expression we used that Now, using the spectral representation (14) we have that Fix x, u and w, and let us focus in (44) on the integral with respect to v. Using the properties of the Fourier transform F with respect to convolution and products, we can write Putting everything in (44), we get Exchanging integrals in (45) yields which is exactly (42). Now, let us focus on the term R d C t (w)e i x,w F [1 Dt(−w) ](y)dw in (42). Using again basic Fourier analysis and Fubini theorem, we obtain which is exactly (43).
Proof. First, call Using polar coordinates, we can write Thanks to (17)- (18) and (25) we have for |x| > 1 We deduce where the last inequality follows from the doubling conditions in [10].
With this estimate, we have that I satisfies: where in the third inequality we used Cauchy-Schwarz inequality and Plancherel theorem.
2 . Using the inequality |ab| ≤ a 2 + b 2 , this yields where in the last inequality we used again doubling conditions in [10].
2,t t 2d . Proof. First, by (43) in Lemma 11, we can write Using Cauchy-Schwarz inequality two times (first with respect to y, then with respect to z) and then Plancherel theorem, II satisfies, with · 2 the L 2 -norm: where the last inequality comes from Young convolution inequalities and the last equality from Plancherel theorem. Since by assumption |F as |x| → ∞, in partic- We deduce, for all t > 0 and all y ∈ R d , that We deduce that where the last equality comes from Plancherel theorem and the last inequality from doubling conditions in [10].
R,t . and the proof is concluded. Now we can proceed with the proof of Proposition 10.
Proof of Proposition 10. First, we assume without loss of generality that w 2,t → ∞, since otherwise the statement follows from theorem 1. Moreover, throughout all the proof we freely use that v 2,t ≍ w 2,t , see (38).
Starting from now, the proof is divided into three steps.
Step 1: Reduction of the proof. We claim that it is enough to check that Y 2,t / Var(Y 2,t ) → N(0, 1) in order to prove Proposition 10. Indeed, since (39) holds and given that d 2 > d−1 2 , we deduce from Lemma 8 that sup r∈R + {r d−1 2 ρ(r)} < ∞. Proposition 4 implies the statement on σ 2 t and justifies that we are left to prove that Y 2,t / Var(Y 2,t ) → N(0, 1).
As done in the proof of Proposition 7, in order to prove the convergence Y 2,t / Var(Y 2,t ) → N(0, 1) we make use of the Fourth Moment Theorem ([17, Theorem 5.2.7]); this requires checking that the only involved contraction goes to zero. To do this, we will use the novel ideas from Fourier analysis introduced in Lemma 11, Lemma 12 and Lemma 13.
As in the proof of Theorem 6, we can first rewrite Y 2,t as a double Wiener-Itô integral with respect to B: with e x such that B x = I 1 (e x ). We know from the Fourth Moment Theorem stated in section 2.4 that Step 2: A two-term error bound for the norm of the contraction. Here we apply Lemma 11. Noting that for x, y ∈ R d either |x − y| ≥ |x|/2 or |y| ≥ |x|/2, we deduce from (42) and (43) the following two terms error bound: Step 3: The norm of the contraction divided by the variance goes to 0. To conclude the proof of Proposition 10, we will now check (see the conclusion of Step 1) that f t ⊗ 1 f t 2 /Var(Y 2,t ) → 0. From (38), we have We deduce from Step 2, Lemma 12 and Lemma 13, that , and the right-hand size goes to 0, since we assumed at the beginning of the proof that w 2,t → ∞.

Proof of theorem 2 when R = 1
This section is devoted to the proof of Theorem 2 in the remaining cases, namely ϕ non-odd, R = 1 and R ′ = 3.
To ease the exposition, we write in the following proposition the statement obtained when, in Theorem 2, we additionally suppose that we are in the cases just mentioned above.  N ∼ N(0, 1). Assume ϕ has Hermite rank R = 1 and let R ′ ≥ 2, R ′ = 3, be its second Hermite rank. Consider Y t defined by (3), where D is compact and Vol(D) > 0. Set m t = E[Y t ] and σ t = Var(Y t ) > 0, and recall the definition (15) of the isotropic spectral measure µ, the definition (12) of Y q,t and the definition (19) of w q,t . Assume that the spectral condition holds as |x| → ∞ 6 , with F the Fourier transform. Then Proof. Set ϕ(x) = ϕ(x) − a 0 − a 1 x. By the very definition of R ′ , the function ϕ has Hermite rank R ′ . Let us define The proof is divided into two steps.
Step 1: CLT for Y t . We claim that σ 2 First of all, observe that (47) implies (20) for R ′ ≥ 2. Moreover, note that we can have three different situations: • If R ′ ≥ 5, then by (47) and Lemma 8 we have sup r∈R + r , and the claim follows immediately by Theorem 1 and the fact that ϕ is not odd.
• If R ′ = 4, then the claim follows by Proposition 7.
as |x| → ∞, then the claim follows by Proposition 10.
Step 2: CLT for Y t . We claim that σ t ∼ σ t as t → ∞ and The proof of Proposition 14 thus follows as soon as these two claims are shown to be true. From (22) we have Then, combining (47), dominated convergence theorem and the fact that as t → ∞. Since in Step 1 we proved that σ 2 t ≍ t d or σ 2 t ≍ t d w R ′ ,t , we have by (49) that σ t ∼ σ t . It remains to prove (51). By orthogonality of chaotic projections we have Since Var(Y 1,t ) = o(t d ), the first addend converges to 0. On the other hand, the second term vanishes because σ t ∼ σ t as t → ∞.
6. An example of application of Theorem 2 In this section, we illustrate a possible use of Theorem 2. In order to introduce our class of fields of interest, recall the definition of the Bessel function J ν given in Section 2.2 and define, for every ν ≥ 0, the normalized Bessel function function ρ ν : [0, ∞) → R as ρ ν (r) = c ν J ν (r) r ν , where c ν is chosen so that ρ ν (0) = 1.
Note that ρd 2 −1 is equal to the function b d defined in (18), and in particular ρ 0 = b 2 = J 0 when d = 2.
Throughout this section, we define the Bessel Gaussian field of order ν and dimension d as the real-valued continuous centered Gaussian field B ν = (B ν x ) x∈R d with covariance function E[B ν x B ν y ] = ρ ν (|x − y|). In particular, d-dimensional Berry's Random Wave Model is defined as the Bessel Gaussian field of order d 2 − 1 and dimension d. The existence of the Bessel Gaussian field of order ν and dimension d ≥ 2 is neither obvious, nor always true. The following result provides a complete picture, and also shows that d-dimensional Berry's Random Wave Model appears to be the critical case for the existence of the Bessel Gaussian field. In this case, the isotropic spectral measure associated to B ν (see (16)) is where c d,ν > 0 is chosen so that µ ν is a probability measure.
Combining this representation with [9, Proposition 2.2], one has that ρ ν is not positive definite when ν < d 2 − 1, showing that the Bessel Gaussian field does not exist in this case.
This means (see (16)) that ρ ν defines a Bessel Gaussian field when ν > d 2 −1 and that its spectral measure is µ ν as defined in the statement. The fact that any Bessel Gaussian field is continuous can be proved using standard results, see e.g. Proof. Using standard asymptotics for Bessel functions (see e.g. [11]), one has ρ ν (r) = c ν cos(r − a ν ) r ν+1/2 + O 1 r ν+3/2 as r → ∞ for some constants c ν and a ν . Then, it follows that Since ν ≥ d 2 − 1, when R ≥ 5 we have that |ρ| q (r)r d−1 is integrable, and the result follows by Breuer Major Theorem 1. For R = 2 and R = 4, the result follows by a direct application of Theorem 2, since the negative moment of order d/R of µ ν exists for every d ≥ 2 and R ≥ 2.
Remark 1. For simplicity, in the previous statement we did not consider all the cases which can be solved applying our Theorem 2. For example, the spectral condition (20) for R = 1 is not verified by µ ν if ν > d 2 − 1, but holds in the d-dimensional Berry case ν = d 2 − 1. Moreover, relying on a slight variation of the proof of Proposition 14, it would have not been very difficult to also cover the non-critical cases where ν ∈ [ d 2 − 1, d 2 − 1 2 ). We skip the details for the sake of brevity.