Mikko S Pakkanen
email: mpakkanen@creates.au.dk

Anthony Réveillac

Functional limit theorems for generalized variations of the fractional Brownian sheet

Keywords: Fractional Brownian sheet, central limit theorem, non-central limit theorem, Hermite sheet, power variation, Malliavin calculus 2010 Mathematics Subject Classification: 60F17 (Primary), 60G60, 60H07 (Secondary)

We prove functional central and non-central limit theorems for generalized variations of the anisotropic d-parameter fractional Brownian sheet (fBs) for any natural number d. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence.

Introduction

Since the seminal works by Breuer and Major [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF], Dobrushin and Major [START_REF] Dobrushin | Non-central limit theorems for non-linear functionals of Gaussian fields[END_REF], Giraitis and Surgailis [START_REF] Giraitis | CLT and other limit theorems for functionals of Gaussian processes[END_REF], Rosenblatt [START_REF] Rosenblatt | Limit theorems for Fourier transforms of functionals of Gaussian sequences[END_REF], and Taqqu [START_REF] Taqqu | Weak Convergence to Fractional Brownian Motion and to the Rosenblatt Process[END_REF][START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence[END_REF][START_REF] Taqqu | A representation for self-similar processes[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF], much attention has been given to the study of the asymptotic behaviour of normalized functionals of Gaussian fields, as these quantities arise naturally in applications, e.g., where models exhibiting longrange dependence are needed. The aforementioned papers focus on nonlinear functionals of a stationary Gaussian field, for which one can derive a central limit theorem (in a finite-dimensional sense or in a functional sense) if the correlation function of the field decays sufficiently fast to zero; see [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF] for a precise formulation. However, if the 1 correlation function decays too slowly to zero, then only a non-central limit theorem can be established, meaning that the limiting distribution fails to be Gaussian; see, e.g., [START_REF] Rosenblatt | Limit theorems for Fourier transforms of functionals of Gaussian sequences[END_REF].

In particular, these results apply to functionals of the fractional Brownian motion (fBm). Let B H := {B H (t) : t ∈ R} be a fBm with Hurst parameter H ∈ (0, 1), which is the unique (in law) H-self similar Gaussian process with stationary increments; see (3.2) and (3.3) below for the definitions of these key properties. The behaviour of the so-called Hermite variations of B H , depending on the value of H, can be described as follows. Let k ∈ {1, 2 . . .} and let P k denote the k-th Hermite polynomial, the definition of which we recall in (2.6) below. Applying results from [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF][START_REF] Dobrushin | Non-central limit theorems for non-linear functionals of Gaussian fields[END_REF][START_REF] Giraitis | CLT and other limit theorems for functionals of Gaussian processes[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF] one obtains that:

(a) If H ∈ 0, 1 -1 2k , then n -1/2 n j=1 P k n H B H j n -B H j-1 n L ---→ n→∞ N 0, σ 2 1 (H, k) . (b) If H = 1 -1 2k , then n log(n) -1/2 n j=1 P k n H B H j n -B H j-1 n L ---→ n→∞ N (0, σ 2 1 (1 -1 2k , k)). (c) If H ∈ 1 -1 2k , 1 , then n 1-2H n j=1 P k n H B H j n -B H j-1 n L 2 (Ω) ----→ n→∞ Hermite 1,k 1 -k(1 -H) .
Above, L -→ denotes convergence in law, N 0, σ 2 1 (H, k) denotes the centered Gaussian law with variance σ 2 1 (H, k) > 0, whereas Hermite 1,k 1 -k(1 -H) stands for a so-called Hermite random variable given by the value of an Hermite process, of order k with Hurst parameter 1 -k(1 -H) ∈ ( 1 2 , 1), at time 1. Such an Hermite process can be represented as a k-fold multiple Wiener integral with respect to Brownian motion, as proven by Taqqu [START_REF] Taqqu | A representation for self-similar processes[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF]. Moreover, the process is non-Gaussian if k ≥ 2. (More details on the Hermite process are provided in §2. [START_REF] Barndorff-Nielsen | Power variation for Gaussian processes with stationary increments[END_REF].) The key observation here is that there are two regimes: Gaussian, subsuming cases (a) and (b), and Hermite, case (c), depending on the Hurst parameter H and on the order k.

The convergences in all cases (a), (b), and (c) can be extended to more general functionals, which we call generalized variations in this paper, obtained by replacing the Hermite polynomial P k with a function

f (u) := ∞ k=k a k P k (u), u ∈ R, (1.1)
where k is the so-called Hermite rank of f . (Naturally, conditions on the summability of the coefficients a k , a k+1 , . . . have to be added.) In this setting, the prevailing regime (Gaussian or Hermite) will depend on the Hurst parameter H and on the Hermite rank k analogously to the simpler setting discussed above. In addition, functional versions of these asymptotic results (under additional assumptions on the coefficients a k , a k+1 , . . .) can be proven in the Skorohod space D([0, 1]); see [START_REF] Taqqu | Weak Convergence to Fractional Brownian Motion and to the Rosenblatt Process[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF].

In connection to applications that involve spatial or spatio-temporal modeling, processes of multiple parameters are also of interest. Recently, there has been interest in understanding the asymptotic behaviour of realized quadratic variations and power variations of ambit fields [START_REF] Barndorff-Nielsen | Volatility determination in an ambit process setting[END_REF][START_REF] Pakkanen | Limit theorems for power variations of ambit fields driven by white noise[END_REF]. An ambit field is an anisotropic multiparameter process driven by white noise, or more generally, by an infinitely-divisible random measure. The problem of finding distributional limits (central or non-central limit theorems) for such power variations is, however, intricate because the dependence structure of an ambit field can be very general; only a "partial" central limit theorem is obtained in [START_REF] Pakkanen | Limit theorems for power variations of ambit fields driven by white noise[END_REF]. As a first approximation, it is thus useful to study this problem with simpler processes that incorporate some of the salient features of ambit fields, such as the non-semimartingality of one-parameter "marginal processes" (see [25, §2.2]) and strong dependence. A tractable process that incorporates some key features of ambit fields is the fractional Brownian sheet (fBs), defined by Ayache et al. [START_REF] Ayache | Drap brownien fractionnaire[END_REF], which is a multi-parameter extension of the fBm. In particular, it is a Gaussian process with stationary rectangular increments.

For concreteness, let Z := {Z(t) : t ∈ [0, 1] 2 } be a two-parameter anisotropic fBs with Hurst parameter (H 1 , H 2 ) ∈ (0, 1) 2 ; see §2.2 for a precise definition. In view of the asymptotic behaviour in cases (a), (b), and (c) involving the fBm, it is natural to ask what is the asymptotic behaviour of Hermite variations of Z with different values of H 1 and H 2 . Consider, for example, the "mixed" case where H 1 < 1 -1 2k and H 2 > 1 -1 2k , which has no counterpart in the one-parameter setting. Because of the structure of the fBs, it is tempting to conjecture that in this case the limiting law is a mixture of a Gaussian law and a marginal law of an Hermite process. However, as shown in [START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF], this is not the case and once again only two limiting laws can be obtained:

(a') If (H 1 , H 2 ) ∈ (0, 1) 2 \ 1 -1 2k , 1 2 , then ϕ(n, H 1 , H 2 ) n j 1 =1 n j 2 =1 P k n H 1 +H 2 Z j 1 -1 n , j 1 n × j 2 -1 n , j 2 n L ---→ n→∞ N 0, σ 2 2 (H 1 , H 2 , k) . (b') If (H 1 , H 2 ) ∈ 1 -1 2k , 1 2 , then ϕ(n, H 1 , H 2 ) n j 1 =1 n j 2 =1 P k n H 1 +H 2 Z j 1 -1 n , j 1 n × j 2 -1 n , j 2 n L 2 (Ω) ----→ n→∞ Hermite 2,k 1 -k(1 -H 1 ), 1 -k(1 -H 2 ) . Above, Z j 1 -1 n , j 1 n × j 2 -1 n , j 2 n
stands for the increment of Z over the rectangle

j 1 -1 n , j 1 n × j 2 -1 n , j 2 n
, defined in §2.3 below, and ϕ(n, H 1 , H 2 ) is a suitable scaling factor; see [29, pp. 9-10] for its definition. The limit in the case (b') is the value of a two-parameter Hermite sheet (see §2.4), of order k with Hurst parameter 1 -k(1 - [START_REF] Ayache | Drap brownien fractionnaire[END_REF][START_REF] Ayache | Drap brownien fractionnaire[END_REF]. Contrary to the one-parameter case, the results obtained in [START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF] are proved only for one-dimensional laws; neither finitedimensional (except in the particular setting of [START_REF] Réveillac | Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets[END_REF]) nor functional convergence (i.e., tightness in a function space) of Hermite variations has been established so far. (In particular in the d-parameter realm with d ≥ 2, tightness is a non-trivial issue, which has not been adressed in [START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF] or in the related paper [START_REF] Réveillac | Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets[END_REF].)

H 1 ), 1 -k(1 -H 2 ) ∈ ( 1 2 , 1) 2 , at point
The first main result of this paper addresses the question about functional convergence in the general, d-parameter case for any d ∈ N. We prove a functional central limit theorem, Theorem 2.11, for generalized variations of a d-parameter anisotropic fBs Z. (As mentioned above, generalized variations extend Hermite variations by replacing P k with a function f of the form (1.1).) This result applies if at least one of the components of the Hurst parameter vector

H = (H 1 , . . . , H d ) ∈ (0, 1) d of Z is less than or equal to 1 -1 2k ,
where k is the Hermite rank of f . A novel feature of this result is that the limiting process is a new fBs, independent of Z, with Hurst parameter vector H = H 1 , . . . , H d given by

H ν := 1 2 , H ν ≤ 1 -1 2k , 1 -k(1 -H ν ), H ν > 1 -1 2k ,
for ν ∈ {1, . . . , d}. Note, in particular, that if H ∈ 0, 1 -1 2k d , then the limit reduces to an ordinary Brownian sheet. The proof of Theorem 2.11 is based on the limit theory for multiple Wiener integrals, due to Nualart and Peccati [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF], and its multivariate extension by Peccati and Tudor [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF]. To prove the functional convergence asserted in Theorem 2.11, we use the tightness criterion of Bickel and Wichura [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF] in the space D([0, 1] d ), which is d-parameter generalization of D([0, 1]), and a moment bound for non-linear functionals of a stationary Gaussian process on Z d (Lemma 4.1).

The second main result of this paper is a functional non-central limit theorem, Theorem 2.18, for generalized variations of Z in the remaining case where each of the components of H is greater than 1 -1 2k . In this case, the limit is a d-parameter Hermite sheet and the convergence holds in probability and also pointwise in L 2 (Ω). Assuming that Z is defined by a moving-average representation with respect to a white noise W on R d , we can give a novel and explicit description of the limit; it is defined using the representation introduced by Clarke De la Cerda and Tudor [START_REF] De La Cerda | Wiener integrals with respect to the Hermite random field and applications to the wave equation[END_REF] with respect to the same white noise W. This makes the relation between Z and the Hermite sheet precise and constitutes a step further compared to the existing literature (see [START_REF] Nourdin | Central and non-central limit theorems for weighted power variations of fractional Brownian motion[END_REF][START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF]), where the limiting Hermite process/sheet is simply obtained as an abstract limit of a Cauchy sequence, from which the properties of the limiting object are deduced.

As an application of Theorems 2.11 and 2.18, we study the asymptotic behaviour of power variations of the fBs Z. As a straightforward consequence of our main results, we obtain a law of large numbers for these power variations. We then study the more delicate question regarding the asymptotic behaviour of rescaled fluctuations of power variations around the limit given by the law of large numbers. In the case of odd power variations, the rescaled fluctuations have a limit, either Gaussian or Hermite, but with even power variations, the fluctuations might not converge in a functional way if d ≥ 2. We show that this convergence issue does not arise at all if one considers instead continuous, multilinear interpolations of power variations. The paper is organized as follows. In §2 we introduce the setting of the paper, some key definitions and the statements of Theorems 2.11 and 2.18. The proofs of these two main results are presented in §3 and §4, the former section collecting the finitedimensional and the latter the functional arguments. Finally, the application to power variations is given in §5.

Preliminaries and main results

Notations

We use the convention that N := {1, 2, . . .} and R + := [0, ∞). The notation |A| stands for the cardinality of a finite set A. For any y ∈ R, we write ⌊y⌋ := max{n ∈ Z : n ≤ v}, {y} := y -⌊y⌋, and (y) + := max(y, 0). The symbol γ denotes the standard Gaussian measure on R, i.e., γ(dy) := (2π) -1/2 exp(-y 2 /2)dy. From now on we fix d in N.

For any vectors s = (s 1 , . . . , s d ) ∈ R d and t = (t 1 , . . . , t d ) ∈ R d , the relation s ≤ t (resp. s < t) signifies that s ν ≤ t ν (resp. s ν < t ν ) for all ν ∈ {1, . . . , d}. We also use the notation

st := (s 1 t 1 , . . . , s d t d ) ∈ R d , s t := s 1 t 1 , . . . , s d t d ∈ R d , ⌊s⌋ := (⌊s 1 ⌋, . . . , ⌊s d ⌋) ∈ Z d , s := s 1 • • • s d ∈
:= [s 1 , t 1 ) × • • • × [s d , t d ) ⊂ R d . Occasionally, we use the norm s ∞ := max(|s 1 |, . . . , |s d |) for s ∈ R d .
For the sake of clarity, we will consistently use the following convention: i, i (1) , i (2) , . . . are multi-indices (vectors) in Z d and j, j 1 , j 2 , . . . are indices (scalars) in Z.

Anisotropic fractional Brownian sheet

We consider an anisotropic, d-parameter fractional Brownian sheet (fBs) Z := {Z(t) : t ∈ R d } with Hurst parameter H ∈ (0, 1) d , which is a centered Gaussian process with covariance

R (d) H (s, t) := E[Z(s)Z(t)] = d ν=1 R (1)
Hν (s ν , t ν ), s, t ∈ R d , (2.1)
where R

(1)

Hν (s ν , t ν ) := 1 2 |s ν | 2Hν + |t ν | 2Hν -|s ν -t ν | 2Hν , s ν , t ν ∈ R,
is the covariance of a fractional Brownian motion with Hurst parameter H ν .

In what follows, it will be convenient to assume that the fBs Z has a particular representation. To this end, let us denote by B 0 (R d ) the family of Borel sets of R d with finite Lebesgue measure. Let (Ω, F, P) be a complete probability space that supports a white noise W := {W(A) :

A ∈ B 0 (R d )}, which is a centered Gaussian process with covariance E[W(A)W(B)] = Leb d (A ∩ B), A, B ∈ B 0 (R d ),
where Leb d (•) denotes the Lebesgue measure on R d . The process Z can be defined as a Wiener integral with respect to W (see, e.g., [START_REF] Nualart | The Malliavin calculus and related topics[END_REF] for the definition), namely

Z(t) := G (d) H (t, u)W(du), t ∈ R d , (2.2)
where the kernel

G (d) H (t, u) := d ν=1 G (1)
Hν (t ν , u ν ), t, u ∈ R d , (2.3)
is defined using the one-dimensional Mandelbrot-Van Ness [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF] kernel

G (1)
Hν (t ν , u ν ) := 1 χ(H ν ) (t ν -u ν ) Hν -1 2 + -(-u ν ) Hν -1 2 + , t ν , u ν ∈ R, (2.4)
with the normalizing constant

χ(H ν ) := 1 2H ν + ∞ 0 (1 + y) Hν -1 2 -y Hν -1 2 dy 1 2
.

We refer to [START_REF] Ayache | Drap brownien fractionnaire[END_REF] for a proof that the process Z defined via (2.2) does indeed have the covariance structure (2.1). The fBs admits a continuous modification (see [3, p. 1040]), so we may assume from now on that Z is continuous.

Increments and generalized variations

Given a function (or a realization of a stochastic process) h : R d → R, we define the increment of h over the half-open hyperrectangle [s, t) ⊂ R d for any s ≤ t by

h([s, t)) := i∈{0,1} d (-1) d-d ν=1 iν g (1 -i)s + it .
This definition can be recovered by differencing iteratively with respect to each of the arguments of the function h. Thus, the increment can be seen as a discrete analogue of the partial derivative

∂ d ∂t 1 •••∂t d .
Remark 2.5. It is useful to note that if there exists functions h ν : R → R, ν ∈ {1, . . . , d}, such that h

(t) = h 1 (t 1 ) • • • h d (t d ) for any t ∈ R d , then h([s, t)) = d ν=1 h ν (t ν ) -h ν (s ν ) ,
which is easily verified by induction with respect to d using iterative differencing.

Let us fix a sequence m(n) n∈N ⊂ N d of multi-indices with the property

m(n) := min m 1 (n), . . . , m d (n) ---→ n→∞ ∞ and a function f ∈ L 2 (R, γ) such that R f (u)γ(du) = 0.
Our aim is to study the asymptotic behaviour of a family U

(n) f

: n ∈ N of d-parameter processes, generalized variations of Z, defined by

U (n) f (t) := 1≤i≤⌊m(n)t⌋ f m(n) H Z i -1 m(n) , i m(n) , t ∈ [0, 1] d , n ∈ N.
The realizations of U

(n) f belong to the space D([0, 1] d ), which for d ≥ 2 is a generalization of the space D([0, 1]) of càdlàg functions on [0, 1]. We refer to [6, pp. 1662] for the definition of the space D([0, 1] d ). In particular, C([0, 1] d ) ⊂ D([0, 1] d ). We endow D([0, 1] d ) with the Skorohod topology described in [6, pp. 1662]. Convergence to a continuous function in this topology is, however, equivalent to uniform convergence (see, e.g., [START_REF] Pakkanen | Limit theorems for power variations of ambit fields driven by white noise[END_REF]Lemma B.2] for a proof in the case d = 2).

Functional limit theorems for generalized variations

We will now formulate two functional limit theorems for the family U (n) f : n ∈ N of generalized variations, defined above. The class of admissible functions f needs to be restricted somewhat, however, and the choice of f and the Hurst parameter H of Z will determine which of the limit theorems applies. Also, we need to rescale U (n) f in suitable way that, likewise, depends on both f and H.

To this end, recall that the Hermite polynomials,

P 0 (u) := 1, P k (u) := (-1) k e u 2 2 d k du k e -u 2 2 , u ∈ R, k ∈ N, (2.6)
form a complete orthogonal system in L 2 (R, γ). Thus, we may expand f in L 2 (R, γ) as

f (u) = ∞ k=k a k P k (u), (2.7)
where the Hermite coefficients a k , a k+1 , . . . ∈ R are such that a k = 0 and

∞ k=k k!a 2 k < ∞. (2.8)
The index k is called the Hermite rank of f , and the proviso R f (u)γ(du) = 0 ensures that k ≥ 1. We will assume that the Hermite coefficients decay somewhat faster than what (2.8) entails.

Assumption 2.9. The Hermite coefficients a k , a k+1 , . . . of the function f satisfy

∞ k=k 3 k 2 √ k!|a k | < ∞.
Let us define a sequence c(n) n∈N ⊂ R d + of rescaling factors by setting for any ν ∈ {1, . . . , d} and n ∈ N,

c ν (n) :=        m ν (n) 2-2k(1-Hν ) , H ν ∈ 1 -1 2k , 1 , m ν (n) log m ν (n) , H ν = 1 -1 2k , m ν (n), H ν ∈ 0, 1 -1 2k . Remark 2.10. Note that lim sup n→∞ mν (n) cν (n) < ∞ and that, in fact, lim n→∞ mν (n) cν (n) = 0 if H ν ∈ 1 -1 2k , 1 .

Now we can define a family U (n)

f : n ∈ N} of rescaled generalized variations as

U (n) f (t) := U (n) f (t) c(n) 1 2 , t ∈ [0, 1] d , n ∈ N.
Our first result is the following functional central limit theorem (FCLT) for generalized variations. Its proof is carried out in §3.2 and §4.2.

Theorem 2.11 (FCLT). Let f be as above such that Assumption 2.9 holds and suppose that H ∈ (0, 1)

d \ 1 -1 2k , 1 d . Then Z, U (n) f L ---→ n→∞ Z, Λ 1 2 H,f Z in D([0, 1] d ) 2 , (2.12)
where Z is a d-parameter fBs with Hurst parameter H ∈ 1 2 , 1 d , independent of Z (defined, possibly, on an extension of (Ω, F, P)), and

Λ H,f := ∞ k=max(k,2) k!a 2 k b (k) ∈ R. (2.

13)

The vectors

H ∈ 1 2 , 1 d and b (k) ∈ R d + , k ≥ max(k, 2
), that appear above are defined by setting for any ν ∈ {1, . . . , d},

H ν := 1 2 , H ν ∈ 0, 1 -1 2k , 1 -k(1 -H ν ), H ν ∈ 1 -1 2k , 1 , (2.14)
and

b (k) ν :=                        j∈Z |j + 1| 2Hν -2|j| 2Hν + |j -1| 2Hν 2 k , H ν ∈ 0, 1 -1 2k , 2 (2k -1)(k -1) 2k 2 k =: ι(k), H ν = 1 -1 2k , k = k, H k ν (2H ν -1) k (1 -k(1 -H ν ))(1 -2k(1 -H ν )) =: κ(H ν , k), H ν ∈ 1 -1 2k , 1 , k = k, 0, H ν ∈ 1 -1 2k , 1 , k > k. Remark 2.15. (1)
The counterpart of the convergence (2.12) for finite dimensional laws holds without Assumption 2.9, see Proposition 3.15, below.

(2) We may use max(k, 2), instead of k, as the lower bound for the summation index k in (2.13) since ι(1) = 0 and

j∈Z |j + 1| 2 Ȟ -2|j| 2 Ȟ + |j -1| 2 Ȟ 2 = j∈Z |j| 2 Ȟ -|j -1| 2 Ȟ 2 - j∈Z |j| 2 Ȟ -|j -1| 2 Ȟ 2 = 0 for any Ȟ ∈ 0, 1 2 . (Then, j∈Z |j| 2 Ȟ -|j-1| 2 Ȟ 2
< ∞ by the mean value theorem.)

(3) The convergence (2.12) can be understood in the framework of stable convergence in law, introduced by Rényi [START_REF] Rényi | On stable sequences of events[END_REF]. Equivalently to (2.12), U

(n)

f converges to Λ 1 2
H,f Z as n → ∞ stably in law with respect to the σ-algebra generated by {Z(t) : t ∈ [0, 1] d }.

(4) See Appendix A for an extension of Theorem 2.11 beyond the present setting with the fBs.

Theorem 2.11 excludes the case H ∈ 1 -1 2k , 1 d . Then, the generalized variations do have a limit, but the limit is non-Gaussian, unless k = 1. To describe the limit, we need the following definition, due to Clarke De la Cerda and Tudor [START_REF] De La Cerda | Wiener integrals with respect to the Hermite random field and applications to the wave equation[END_REF].

Definition 2.16. An anisotropic, d-parameter Hermite sheet Z := Z(t) : t ∈ R d + of order k ≥ 2 with Hurst parameter H ∈ 1 2 , 1 d is defined as a k-fold multiple Wiener integral (see §3.
2) with respect to the white noise W,

Z(t) := • • • G (k) H t, u (1) , . . . , u (k) W du (1) • • • W du (k) := I W k G (k) H t, • (2.17)
for any t ∈ R d + . In (2.17), the kernel G

(d,k) H (t, •) ∈ L 2 (R kd ) is given by G (k) H t, u (1) , . . . , u (k) := 1 χ H, k [0,t) k κ=1 d ν=1 y ν -u (κ) ν -1 2 -1-Hν k + dy, u (1) , . . . , u (k) ∈ R d , using the normalizing constant χ H, k := d ν=1 B 1 2 -1-Hν k , 2( Hν -1) k H ν (2 H ν -1) 1 2
, where B stands for the beta function.

The Hermite sheet Z is self-similar and has the same correlation structure as a fBs with Hurst parameter H. In the case k = 1, the process Z is Gaussian (in fact, it coincides with a fractional Brownian sheet with Hurst parameter H) but for k ≥ 2 it is non-Gaussian. In the case k = 2, the name Rosenblatt sheet (and Rosenblatt process, when d = 1; see [START_REF] Tudor | Analysis of the Rosenblatt process[END_REF]) is often used, in honor of Murray Rosenblatt's seminal paper [START_REF] Rosenblatt | Independence and dependence[END_REF]. See also the recent papers [START_REF] Maejima | On the distribution of the Rosenblatt process[END_REF][START_REF] Veillette | Properties and numerical evaluation of the Rosenblatt distribution[END_REF] for more details on the Rosenblatt distribution, including proofs that this distribution is infinitely divisible.

As our second main result, we obtain the following functional non-central limit theorem (FNCLT) for generalized variations. The proof of this result is carried out in §3.3 and §4.2.

Theorem 2.18 (FNCLT). Let f be as above such that Assumption 2.9 holds and suppose that

H ∈ 1 -1 2k , 1 d . Then, U (n) f P ---→ n→∞ Λ 1 2 H,f Z in D([0, 1] d ), (2.19)
where Z is a d-parameter Hermite sheet of order k with Hurst parameter H, given by (2.14), and Λ H,f is given by (2.13).

Remark 2.20. (1) The convergence (2.19) holds pointwise in L 2 (Ω, F, P), even when Assumption 2.9 does not hold, see Proposition 3.26, below.

(2) Unlike in Theorem 2.11, the non-central limit Z is defined on the original probability space (Ω, F, P). In particular, Z is driven by the same white noise W as Z.

(3) In the special case k = 1, the limit in (2.19) is Gaussian. In fact, then Λ H,f = a 2 1 and Z = Z.

Remark 2.21. Our method of proving the convergence of finite-dimensional distributions of U

(n)

f , using chaotic expansions, is particularly suitable for providing estimates on the speed of convergence (for example in the Wasserstein distance) as is done in [START_REF] Nourdin | Multivariate normal approximation using Stein's method and Malliavin calculus[END_REF] following the original idea presented in [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF], which combines the Malliavin calculus and Stein's method. In addition, the study of weighted variations of the fBs is still partially incomplete, especially with regards to functional convergence (see [START_REF] Réveillac | Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets[END_REF]). To keep the length of this paper within limits -and since proving functional convergence of weighted variations requires slighly different methods -we have decided to treat these two questions in a separate paper.

Finite-dimensional convergence

In this section, we begin the proofs of Theorems 2.11 and 2.18. To be more precise, we prove the finite-dimensional statements corresponding to (2.12) and (2.19), see Propositions 3.15 and 3.26, respectively. As a preparation, we study the correlation structure of the increments of the fBs Z and recall the chaotic expansion of functionals of Z.

Correlation structure of increments

In what follows, it will be convenient to use the shorthand

Z (n) i := m(n) H Z i -1 m(n) , i m(n) , 1 ≤ i ≤ m(n), n ∈ N. (3.1)
For any n ∈ N, the family

Z (n) i : 1 ≤ i ≤ m(n)
is clearly centered and Gaussian. We will next derive its correlation structure.

To describe the correlation structure of the rescaled increments (3.1), let {B Ȟ (t) : t ∈ R} be an auxiliary fractional Brownian motion with Hurst parameter Ȟ ∈ (0, 1). Using the kernel (2.4), we may represent it as

B Ȟ (t) := R G (1) Ȟ (t, u)dB(u), t ∈ R, where {B(t) : t ∈ R} is a standard Brownian motion. Recall that B Ȟ is Ȟ-self similar, i.e., B Ȟ (at) : t ∈ R L = a Ȟ B Ȟ (t) : t ∈ R for any a > 0, (3.2)
and has stationary increments, i.e.,

B Ȟ ([s, s + t)) : t ∈ R L = {B Ȟ ([0, t)) : t ∈ R} for any s ∈ R. (3.3)
The discrete parameter process

B Ȟ ([j, j + 1)), j ∈ Z,
which is stationary by (3.3), is called a fractional Gaussian noise. Its correlation function can be expressed as

r Ȟ (j) := E B Ȟ ([j, j + 1))B Ȟ ([0, 1)) = |j + 1| 2 Ȟ -2|j| 2 Ȟ + |j -1| 2 Ȟ 2 , j ∈ Z.
One can show, e.g., using the mean value theorem, that there exists a constant

C( Ȟ) > 0 such that |r Ȟ (j)| ≤ C( Ȟ)|j| -2(1-Ȟ) , j ∈ Z. (3.4) Thus, if k > 1 2 and Ȟ ∈ 0, 1 -1 2k , then j∈Z |r Ȟ (j)| k < ∞. (3.5) If Ȟ ∈ 1 -1 2k
, 1 , then the series (3.5) diverges. In this case it is still useful to have estimates for the partial sums corresponding to (3.5). Using (3.4), one can prove that there exists a constant C ′ ( Ȟ, k) > 0 such that

|j|<l |r Ȟ (j)| k ≤ C ′ ( Ȟ, k) log l, Ȟ = 1 -1 2k , C ′ ( Ȟ, k)l 1-2k(1-Ȟ) , Ȟ ∈ 1 -1 2k , 1 . (3.6)
We can now describe the correlations of the rescaled increments (3.1) using the correlation function of the fractional Gaussian noise as follows.

Lemma 3.7 (Correlation structure). For any n ∈ N, and

1 ≤ i (1) , i (2) ≤ m(n), E Z (n) i (1) Z (n) i (2) = d ν=1 r Hν i (1) ν -i (2) ν .
Proof. Using first the linearity of Wiener integrals and then the product structure (2.3) of the kernel

G (d)
H and Remark 2.5, we obtain for any s, t

∈ [0, 1] d such that s ≤ t, Z([s, t)) = G (d) H ([s, t), u)W(du) = d ν=1 G (1)
Hν ([s ν , t ν ), u ν )W(du). (3.8)
Thus, by Fubini's theorem,

E Z i (1) -1 m(n) , i (1) m(n) Z i (2) -1 m(n) , i (2) m(n) = d ν=1 G (1) Hν i (1) ν -1 m ν (n) , i (1)
ν m ν (n) , v G (1) Hν i (2) ν -1 m ν (n) , i (2)
ν m ν (n) , v dv = d ν=1 E B Hν i (1) ν -1 m ν (n) , i (1)
ν m ν (n) B Hν i (2) ν -1 m ν (n) , i (2)
ν m ν (n) .
For any ν ∈ {1, . . . , d}, the fractional Brownian motion B Hν is H ν -self similar and has stationary increments, cf. (3.2) and (3.3), so we obtain

E B Hν i (1) ν -1 m ν (n) , i (1)
ν m ν (n) B Hν i (2) ν -1 m ν (n) , i (2)
ν m ν (n) = r Hν i (1)
ν -i (2) ν m ν (n) 2Hν ,
from which the assertion follows.

Multiple Wiener integrals and central limit theorem

The proofs of Theorems 2.11 and 2.18 rely on particular representations of generalized variations in terms of multiple Wiener integrals with respect to the underlying white noise W. We will now briefly review the theory of multiple Wiener integrals and how these integrals can be used to prove central limit theorems. As an application, we take the first step in the proof of Theorem 2.11 by establishing the convergence of finitedimensional laws.

In what follows, we write

H := L 2 (R d ).
Recall that H is a separable Hilbert space when we endow it with the usual inner product. For any k ∈ N, we denote by H ⊗k the k-fold tensor power of H and by H ⊙k ⊂ H ⊗k the symmetrization of H ⊗k . Note that we can make the identification H ⊗k ∼ = L 2 (R kd ). For any h ∈ H ⊙k , we may define the k-fold multiple Wiener integral I W k (h) of h with respect to W. This is done, using Hermite polynomials, by setting for any κ ∈ N, any orthonormal h 1 , . . . , h κ ∈ H, and for any k

1 , . . . , k κ ∈ N such that k 1 + • • • + k κ = k, I W k κ j=1 h ⊗k j j := k! κ j=1 P k j h j (u)W(du) , (3.9)
where ⊙ denotes symmetrization of the tensor product, and extended to general integrands h ∈ H ⊙k using a density argument. It is worth stressing that the multiple Wiener integral is linear with respect to the integrand and has zero expectation. Moreover, by (3.9), for h ∈ H one has

I W 1 (h) = h(u)W(du), (3.10)
and if h H = 1, then for any k ∈ N it holds that h ⊗k ∈ H ⊙k and

P k I W 1 (h) = I W k (h ⊗k ). (3.11)
Multiple Wiener integrals have the following isometry and orthogonality properties: for any

k 1 , k 2 ∈ N, h 1 ∈ H ⊙k 1 , and h 2 ∈ H ⊙k 2 , E I W k 1 (h 1 )I W k 2 (h 2 ) = k 1 ! h 1 , h 2 H ⊗k 1 , k 1 = k 2 , 0, k 1 = k 2 . (3.12)
Recall that any random variable Y ∈ L 2 (Ω, F, P) has a chaotic expansion in terms of kernels

F Y k ∈ H ⊙k , k ∈ N, as Y = E[Y ] + ∞ k=1 I W k F Y k , (3.13)
where the series converges in L 2 (Ω, F, P) (see, e.g, [START_REF] Kallenberg | Foundations of modern probability[END_REF]Theorem 13.26]). Since the apperance of the seminal paper of Nualart and Peccati [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF], the convergence of random variables admitting expansions of the form (3.13) to a Gaussian law has been well understood, based on convenient characterizations using the properties of the kernels.

To describe the key result, recall that for any k 1 , k 2 , r ∈ N such the r < min{k 1 , k 2 }, the r-th contraction of h 1 ∈ H ⊗k 1 and h 2 ∈ H ⊗k 2 is defined as

(h 1 ⊗ r h 2 ) t (1) , . . . , t (k 1 +k 2 -2r) := h 1 t (1) , . . . , t (k 1 -r) , • , h 2 • , t (k 1 -r+1) , . . . , t (k 1 +k 2 -2r)
H ⊗r for any t (1) , . . . , t (k 1 +k 2 -2r) ∈ R d . The following multivariate central limit theorem for chaotic expansions appears in [4, Theorem 5], where it is proven using the results in [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF].

Lemma 3.14 (CLT for chaotic expansions). Let κ ∈ N and suppose that for any n ∈ N, we are given random variables

Y (n) 1 , . . . , Y (n) κ
∈ L 2 (Ω, F, P) such that for any j ∈ {1, . . . , κ},

Y (n) j = ∞ k=1 I W k F (n) k (j, •) ,
where

F (n) k (j, •) ∈ H ⊙k , k ∈ N.
Let us assume that the following conditions hold:

(a) For any j ∈ {1, . . . , κ},

lim sup n→∞ ∞ k=K k! F (n) k (j, •) 2 H ⊗k ----→ K→∞ 0.
(b) There exists a sequence Σ, Σ 1 , Σ 2 , . . . of positive semidefinite d × d-matrices such that for any (j 1 , j 2 ) ∈ {1, . . . , κ} 2 and k ∈ N,

k! F (n) k (j 1 , •), F (n) k (j 2 , •) H ⊗k ---→ n→∞ Σ k (j 1 , j 2 ),
and that ∞ k=1 Σ k = Σ.

(c) For any j ∈ {1, . . . , κ}, k ≥ 2, and r ∈ {1, . . . , k -1},

F (n) k (j, •) ⊗ r F (n) k (j, •) 2 H ⊗2(k-r) ---→ n→∞ 0.
Then we have

Y (n) 1 , . . . , Y (n) κ L ---→ n→∞ N κ (0, Σ),
where N κ (0, Σ) stands for the κ-dimensional Gaussian law with mean 0 and covariance matrix Σ.

We apply now Lemma 3.14 to establish the following finite-dimensional version of Theorem 2.11. Proposition 3.15 (CLT for finite-dimensional laws). Suppose that H ∈ (0, 1)

d \ 1 - 1 2k , 1 d . Let κ ∈ N and t (1) , . . . , t (κ) ∈ [0, 1] d κ . Then, Z(t (1) ), . . . , Z(t (κ) ), U (n) f (t (1) ), . . . , U (n) f (t (κ) ) L ---→ n→∞ N 2κ 0, Ξ 0 0 Σ , (3.16)
where Ξ is the covariance matrix of the random vector Z(t (1) ), . . . , Z(t (κ) ) and

Σ(j 1 , j 2 ) := Λ H,f R (d) H
(t (j 1 ) , t (j 2 ) ), (j 1 , j 2 ) ∈ {1, . . . , κ} 2 .

Remark 3.17. In the case H ∈ 0, 1 -1 2k d , the convergence

U (n) f (t (1) ), . . . , U (n) f (t (κ) ) L ---→ n→∞ N κ (Σ)
follows from the classical results of Breuer and Major [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF].

Proof of Proposition 3.15. By (3.10), we have

Z(t) = I W 1 G (d)
H (t, •) for any t ∈ [0, 1] d . In particular, by (3.8) and linearity, we find that for any n ∈ N and 1

≤ i ≤ m(n), Z (n) i = I W 1 h (n) i
, where

h (n) i := m(n) H g (n) i , g (n)
i := G (d) H i -1 m(n) , i m(n) , • , satisfying h (n) i H = 1,
U (n) f (t) = ∞ k=k I W k F (n) k (t, •) , t ∈ [0, 1] d , n ∈ N, (3.18)
where

F (n) k (t, •) := a k c(n) 1/2 1≤i≤⌊m(n)t⌋ h (n) i ⊗k , k ≥ k. (3.19)
For the remainder of the proof, let s, t ∈ {t (1) , . . . , t (κ) }. Let us first look into condition (a) of Lemma 3.14. By Lemma 3.7 and the relation (3.12), we obtain for any n ∈ N and k ≥ k,

F (n) k (s, •), F (n) k (t, •) H ⊗k = a 2 k c(n) 1≤i (1) ≤⌊m(n)s⌋ 1≤i (2) ≤⌊m(n)t⌋ h (n) i (1) ⊗k , h (n) i (2) ⊗k H ⊗k = a 2 k c(n) 1≤i (1) ≤⌊m(n)s⌋ 1≤i (2) ≤⌊m(n)t⌋ h (n) i (1) , h (n) i (2) k H = a 2 k d ν=1 1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)tν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k . (3.20)
Let k 0 ∈ N be large enough so that H ν ∈ 0, 1 -1 2k 0 for any ν ∈ {1, . . . , d}. Then we have for any

k ≥ k 0 , 0 ≤ d ν=1 1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)sν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k ≤ d ν=1 1 c ν (n) mν (n) j 1 =1 mν (n) j 2 =1 |r Hν (j 1 -j 2 )| k 0 ≤ d ν=1 sup n∈N m ν (n) c ν (n) j∈Z |r Hν (j)| k 0 < ∞,
which follows from Remark 2.10 and the elementary estimate sup

1≤j 1 ≤ℓ ℓ j 2 =1 |r Hν (j 1 -j 2 )| q ≤ |j|<ℓ |r Hν (j)| q , ℓ ∈ N, q ∈ R + . (3.21)
Thus, by (2.8), we have for

K ≥ k 0 , 0 ≤ lim sup n→∞ ∞ k=K k! F (n) k (s) 2 H ⊗k ≤ ∞ k=K k!a 2 k d ν=1 sup n∈N m ν (n) c ν (n) j∈Z |r Hν (j)| k 0 ----→ K→∞ 0,
and the condition (a) is verified.

To check condition (b) of Lemma 3.14, note that we can write for any ν ∈ {1, . . . , d}, assuming without loss of generality that t ν ≥ s ν ,

1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)tν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k = 1 2 1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)sν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k + 1 c ν (n) ⌊mν (n)tν ⌋ j 1 =1 ⌊mν (n)tν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k - 1 c ν (n) ⌊mν (n)tν ⌋-⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)tν ⌋-⌊mν (n)sν ⌋ j 2 =1
r Hν (j 1 -j 2 ) k .

(3.22)

We will now compute the limit of (3.22) separately in the following three possible cases:

(i) H ν ∈ 1 -1 2k , 1 , (ii) H ν = 1 -1 2k , (iii) H ν ∈ 0, 1 -1 2k .
In the case (i), we obtain, by Lemma A.1 of [START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF],

1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)sν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k = ⌊m ν (n)s ν ⌋ m ν (n) 2-2k(1-Hν ) ⌊m ν (n)s ν ⌋ -2+2k(1-Hν ) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)sν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k ---→ n→∞ κ(H ν , k)s 2-2k(1-Hν ) ν = κ(H ν , k)s Hν ν .
(In fact, Lemma A.1 of [START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF] requires that k ≥ 2, but it is straightforward to check that the limits stated therein are valid also when k = 1.) With k > k we may choose ε > 0 so that k + ε < min

1 2(1-Hν ) , k , whence 1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)sν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k ≤ 1 c ν (n) mν (n) j 1 =1 mν (n) j 2 =1 |r Hν (j 1 -j 2 )| k+ε ≤ 1 m ν (n) 1-2k(1-Hν ) |j|<mν (n) |r Hν (j)| k+ε ---→ n→∞ 0 (3.23)
by the estimate (3.6). Treating the other summands on right-hand side of (3.22) similarly, we arrive at

lim n→∞ 1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)tν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k =    κ(Hν ,k) 2 s Hν ν + t Hν ν -(t ν -s ν ) Hν = κ(H ν , k)R (1) Hν (s ν , t ν ), k = k, 0, k > k.
In the case (ii), rearranging and applying Lemma A.1 of [START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF] yields

1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)sν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k = 1 + log ⌊mν (n)sν ⌋ mν (n) log m ν (n) ⌊m ν (n)s ν ⌋/m ν (n) ⌊m ν (n)s ν ⌋ log(⌊m ν (n)s ν ⌋) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)sν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k ---→ n→∞ ι(k)s ν .
When k > k, we have H ν ∈ 0, 1 -1 2k and, consequently,

1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)sν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k ≤ 1 log m ν (n) j∈Z |r Hν (j)| k ---→ n→∞ 0.
Again, a similar treatment of the other summands on right-hand side of (3.22) establishes that

lim n→∞ 1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)tν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k =    ι(k) 2 s ν + t ν -(t ν -s ν ) = ι(k)R (1) Hν (s ν , t ν ), k = k, 0, k > k.
Finally, in the case (iii), we deduce in a straightforward manner that for any k ≥ k,

lim n→∞ 1 c ν (n) ⌊mν (n)sν ⌋ j 1 =1 ⌊mν (n)tν ⌋ j 2 =1 r Hν (j 1 -j 2 ) k = 1 2 j∈Z r Hν (j) k s ν + t ν -(t ν -s ν ) = j∈Z r Hν (j) k R (1) Hν
(s ν , t ν ) using Lemma A.1 of [START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF].

Returning to the expression (3.20), we have shown that for any

k ≥ k, k! F (n) k (s), F (n) k (t) H ⊗k ---→ n→∞ k!a 2 k b (k) R (d) H (s, t). (3.24)
When k = 1, we need to check, additionally, that the covariance matrix appearing in the limit (3.16) is block-diagonal. To this end, note that it follows from the assumption

H ∈ (0, 1) d \ 1 2 , 1 d , that b (1)
ν = 0 for some ν ∈ {1, . . . , d}, which in turn implies that

F (n) 1 (s, •) 2 H ---→ n→∞ 0.
By the Cauchy-Schwarz inequality, we have then

F (n) 1 (s, •), G (d)
H (t, •) H ---→ n→∞ 0,
which ensures block diagonality, and concludes the verification of condition (b).

In order to check condition (c) of Lemma 3.14, let k ≥ max(k, 2) and r ∈ {1, . . . , k -1}. Using the bilinearity of contractions and inner products, we obtain

F (n) k (t, •) ⊗ r F (n) k (t, •) 2 H ⊗2(k-r) = a 4 k c(n) 2 1≤i (j) ≤⌊m(n)t⌋ j∈{1,2,3,4} h (n) i (1) ⊗k ⊗ r h (n) i (2)
⊗k , h

(n) i (3) ⊗k ⊗ r h (n) i (4) ⊗k H ⊗2(k-r) = a 4 k c(n) 2 1≤i (j) ≤⌊m(n)t⌋ j∈{1,2,3,4} h (n) i (1) , h (n) i (2) r H h (n) i (3) , h (n) i (4) r H h (n) i (1) , h (n) i (3) k-r H h (n) i (2) , h (n) i (4) k-r H = a 4 k d ν=1 1 c ν (n) 2 ⌊mν (n)tν ⌋ j 1 ,j 2 ,j 3 ,j 4 =1
r Hν (j 1 -j 2 ) r r Hν (j 3 -j 4 ) r r Hν (j 1 -j 3 ) k-r r Hν (j 2 -j 4 ) k-r .

Following the proof of Lemma 4.1 of [START_REF] Nourdin | Quantitative Breuer-Major theorems[END_REF], we apply the bound

|r Hν (j 1 )| r |r Hν (j 2 )| k-r ≤ |r Hν (j 1 )| k + |r Hν (j 2 )| k , j 1 , j 2 ∈ Z,
which is a consequence of Young's inequality, and use repeatedly (3.21) to deduce that

F (n) k (t, •) ⊗ r F (n) k (t, •) 2 H ⊗2(k-r) ≤ 16 d a 4 k d ν=1 m ν (n)φ ν (n) c ν (n) 2 , (3.25)
where

φ ν (n) := |j 1 |<mν (n) |r Hν (j 1 )| r |j 2 |<mν (n) |r Hν (j 2 )| k-r |j 3 |<mν (n)
|r Hν (j 3 )| k .

We need to analyze the asymptotic behaviour of φ ν (n) as n → ∞. This can be accomplished by considering separately the three possible cases:

(i') H ν ∈ 1 -1 2k , 1 , (ii') H ν = 1 -1 2k , (iii') H ν ∈ 0, 1 -1 2k . In the case (i') we have clearly H ν ∈ 1 -1 2(k-r) , 1 ∩ 1 -1 2r
, 1 , and by the estimate (3.6), it follows that

φ ν (n) ≤ C ′′ (H ν , k, r)m ν (n) 3-4k(1-Hν ) , where C ′′ (H ν , k, r) := C ′ (H ν , r)C ′ (H ν , k -r)C ′ (H ν , k). Since H ν ∈ 1 -1 2k , 1 ⊂ 1 -1 2k , 1 , we obtain lim sup n→∞ m ν (n)φ ν (n) c ν (n) 2 ≤ lim sup n→∞ C ′′ (H ν , k, r) m ν (n) 4(k-k)(1-Hν ) < ∞.
Let us then consider to the case (ii'). We have still

H ν ∈ 1 -1 2(k-r) , 1 ∩ 1 -1 2r , 1 , so by (3.6) we find that φ ν (n) ≤ C ′′ (H ν , k, r)m ν (n) 2-2k(1-Hν ) log m ν (n) = C ′′ (H ν , k, r)m ν (n) log m ν (n) . Necessarily H ν ∈ 1 -1 2k , 1 , whence there is an index n 0 ∈ N such that c ν (n) ≥ m ν (n) log m ν (n) for all n ≥ n 0 . We deduce then that lim n→∞ m ν (n)φ ν (n) c ν (n) 2 ≤ lim n→∞ C ′′ (H ν , k, r) log m ν (n) = 0.
In the remaining case (iii') we have j∈Z |r Hν (j [START_REF] Nourdin | Quantitative Breuer-Major theorems[END_REF].

)| k < ∞. Since there is n 0 ∈ N such that c ν (n) ≥ m ν (n) for all n ≥ n 0 , we find that lim n→∞ m ν (n)φ ν (n) c ν (n) 2 ≤ lim n→∞ 1 m ν (n) |j 1 |<mν (n) |r Hν (j 1 )| r |j 2 |<mν (n) |r Hν (j 2 )| k-r j 3 ∈Z |r Hν (j 3 )| k = 0 by Lemma 2.2 of
Finally, let us return to the upper bound (3.25). The crucial observation is that the assumption H ∈ (0, 1) d \ 1 -1 2k , 1 d implies that there is at least one coordinate ν ∈ {1, . . . , d} that falls within case (ii') or (iii'). Thus,

F (n) k (t, •) ⊗ r F (n) k (t, •) 2 H ⊗2(k-r) ---→ n→∞ 0,
concluding the verification of the condition (c), and the convergence (3.16) follows.

Convergence to the Hermite sheet

We prove next a pointwise version of Theorem 2.18 in L 2 (Ω). The argument is based mainly on the chaotic expansion (3.18) and the isometry property (3.12) of multiple Wiener integrals. However, compared to the proof of Proposition 3.15, we need to analyze the asymptotic behaviour of the associated kernels more carefully.

Proposition 3.26 (Pointwise NCLT). Suppose that H ∈ 1 -1 2k , 1 d . Then, for any

t ∈ [0, 1] d , U (n) f (t) L 2 (Ω) ----→ n→∞ Λ 1 2 H,f Z(t), (3.27)
where Z is the Hermite sheet appearing in Theorem 2.18.

Proof. Fix t ∈ [0, 1] d . By the chaotic expansion (3.18), we have for any n ∈ N,

U (n) f (t) = I W k F (n) k (t, •) + ∞ k=k+1 I W k F (n) k (t, •) .
Using the property (3.12) and Parseval's identity, we find that

E ∞ k=k+1 I W k F (n) k (t, •) 2 = ∞ k=k+1 k! F (n) k (t, •) 2 H ⊗k .
Since H ∈ 1-1 2k , 1 d , we may choose ε ∈ (0, 1] so that H ∈ 1-

∞ k=k+1 k! F (n) k (t, •) 2 H ⊗k ≤ ∞ k=k+1 k!a 2 k d ν=1 1 m ν (n) 1-2k(1-Hν ) |j|<mν (n) |r Hν (j)| k+ε ---→ n→∞ 0,
where convergence to zero is a consequence of the bound (3.6). Thus, it remains to show that

I W k F (n) k (t, •) L 2 (Ω) ----→ n→∞ I W k Λ 1 2 H,f G (k) H (t, •) = Λ 1 2 H,f Z(t),
which follows by (3.12), if we can show that

F (n) k (t, •) H ⊗k ---→ n→∞ Λ 1 2 H,f G (k) H (t, •). (3.28)
In the special case k = 1, the convergence (3.27) follows already. Namely,

I W 1 F (n) 1 (t, •) = a 1 Z ⌊m(n)t⌋ m(n) L 2 (Ω) ----→ n→∞ a 1 Z(t) = Λ 1 2 H,f Z(t) = Λ 1 2 H,f Z(t)
by the L 2 -continuity of Z. Thus, we can assume that k ≥ 2 from now on.

We will prove the convergence (3.28) in two steps. First, we show that

F (n) k (t,
•) n∈N is a Cauchy sequence in H ⊗k . Later, we characterize the limit. Let n 1 , n 2 ∈ N and consider

F (n 1 ) k (t, •) -F (n 2 ) k (t, •) 2 H ⊗k = F (n 1 ) k (t, •) 2 H ⊗k + F (n 2 ) k (t, •) 2 H ⊗k -2 F (n 1 ) k (t, •), F (n 2 ) k (t, •) H ⊗k . (3.29)
By Definition (3.19), we have

F (n 1 ) k (t, •), F (n 2 ) k (t, •) H ⊗k = a 2 k m(n 1 ) k-1 m(n 2 ) k-1 1≤i (1) ≤⌊m(n 1 )t⌋ 1≤i ≤⌊m(n 2 )t⌋ g (n 1 ) i (1) , g (n 2 ) i (2) k H ⊗k .
Mimicking the proof of Lemma 3.7, we obtain

g (n 1 ) i (1) , g (n 2 ) i (2) H ⊗k = d ν=1 G (1) Hν i (1) ν -1 m ν (n 1 ) , i (1)
ν m ν (n 1 ) , v G (1) Hν i (2) ν -1 m ν (n 2 ) , i (2) ν m ν (n 2 ) , v dv = d ν=1 E B Hν i (1) ν -1 m ν (n 1 ) , i (1)
ν m ν (n 1 ) B Hν i (2) ν -1 m ν (n 2 ) , i (2)
ν m ν (n 2 ) = d ν=1 H ν (2H ν -1) i (1) ν mν (n 1 ) i (1) ν -1 mν (n 1 ) i (2) ν mν (n 2 ) i (2) ν -1 mν (n 2 ) |v 1 -v 2 | -2(1-Hν ) dv 1 dv 2 ,
where the final equality follows (see, e.g., [17, p. 574]) since H ν > 1 -1 2k > 1 2 for any ν ∈ {1, . . . , d}. Adapting the argument used in [18, pp. 1064-1065], we deduce that lim

n 1 ,n 2 →∞ F (n 1 ) k (t, •), F (n 2 ) k (t, •) H ⊗k = a 2 k d ν=1 H k ν (2H ν -1) k t 0 t 0 |v 1 -v 2 | -2k(1-Hν ) dv 1 dv 2 = a 2 k d ν=1 t 2 Hν ν H k ν (2H ν -1) k 1 0 1 0 |v 1 -v 2 | -2k(1-Hν ) dv 1 dv 2 = a 2 k d ν=1 t 2 Hν ν κ(H ν , k) = a 2 k b (k) R (d) H (t, t).
(3.30) Thus, by (3.24) and (3.29), lim

n 1 ,n 2 →∞ F (n 1 ) k (t, •) -F (n 2 ) k (t, •) 2
H ⊗k = 0, whence F (n) k (t, •) n∈N is a Cauchy sequence.
To characterize the limit of

F (n) k (t, •) n∈N , let us consider for any s (1) , . . . , s (k) ∈ R d , F (n) k t, s (1) , . . . , s (k) = a k m(n) k-1 1≤i≤⌊m(n)t⌋ k κ=1 G (d) H i -1 m(n) , i m(n) , s (κ) = a k m(n) k-1 1≤i≤⌊m(n)t⌋ k κ=1 d ν=1 G (1) Hν i ν -1 m ν (n) , i ν m ν (n) , s (κ) ν = a k d ν=1 1 m ν (n) ⌊mν (n)tν ⌋ j=1 k κ=1 m ν (n)G (1) Hν j -1 m ν (n) , j m ν (n) , s (κ) ν ,
where the second equality is a consequence of Remark 2.5. Since

G (1) Hν j -1 m ν (n) , j m ν (n) , s (κ) ν = 1 χ(H ν ) j m ν (n) -s (κ) ν Hν -1 2 + - j -1 m ν (n) -s (κ) ν Hν -1 2 +
, it follows from Lemma 3.32, below, that

F (n) k (t, •) ---→ n→∞ C ′′′ a k , H, k G (k) H (t, •) a.e. on R kd (3.31)
for some constant C ′′′ a k , H, k > 0. By the Cauchy property of F

that E Z(t) 2 = k! G (k) H (t, •) 2 H ⊗k = R (d) H
(t, t). In view of (3.30), we find that

C ′′′ a k , H, k 2 = k!a 2 k b (k) = Λ H,f , whence (3.28) follows.
The following technical lemma was essential in the proof of Proposition 3.26.

Lemma 3.32. Suppose that k ≥ 2, Ȟ ∈ 1 2 , 1 , and v > 0. Then,

1 n ⌊nv⌋ j=1 k κ=1 n j n -s κ Ȟ-1 2 + - j -1 n -s κ Ȟ-1 2 + ---→ n→∞ Ȟ - 1 2 k v 0 k κ=1 (u-s κ ) Ȟ-3 2 + du (3.33) for almost any s = (s 1 , . . . , s k ) ∈ R k .
Proof. We may assume that s := max(s 1 , . . . , s k ) < v, as otherwise (3.33) is trivially true. In fact,

v 0 k κ=1 (y -s κ ) Ȟ-3 2 + dy = v s k κ=1 (y -s κ ) Ȟ-3 2 dy.
We split the sum on the left-hand side of (3.33) for any n ∈ N, such that ⌊nv⌋ > ⌊ns⌋+3, as

1 n ⌊nv⌋ j=1 k κ=1 n j n -s κ Ȟ-1 2 + - j -1 n -s κ Ȟ-1 2 + = 1 n ⌊ns⌋+2 j=⌊ns⌋+1 k κ=1 n j n -s κ Ȟ-1 2 - j -1 n -s κ Ȟ-1 2 + + 1 n ⌊nv⌋ j=⌊ns⌋+3 k κ=1 n j n -s κ Ȟ-1 2 - j -1 n -s κ Ȟ-1 2 =: S (1) n + S (2)
n . Using the mean value theorem, we obtain for any y ∈ R and n, j ∈ N, such that j-1 n > y, the bounds

n j n -y Ȟ-1 2 - j -1 n -y Ȟ-1 2 ≤ Ȟ - 1 2 j -1 n -y Ȟ-3 2 , (3.34)
n j n -y Ȟ-1 2 - j -1 n -y Ȟ-1 2 ≥ Ȟ - 1 2 j n -y Ȟ-3 2 . (3.35)
Since we are aiming to prove (3.33) for almost any s ∈ R k , we may assume (by symmetry) that s = s 1 > s κ for any κ ∈ {2, . . . , k}. Then we have for j ∈ {1, 2}, lim sup

n→∞ k κ=2 n ⌊ns⌋ + j n -s κ Ȟ-1 2 - ⌊ns⌋ + j -1 n -s κ Ȟ-1 2 + < ∞
by (3.34), and

0 ≤ ⌊ns⌋ + j n -s 1 Ȟ-1 2 - ⌊ns⌋ + j -1 n -s 1 Ȟ-1 2 + ≤ ⌊ns⌋ + 2 n -s 1 Ȟ-1 2 ---→ n→∞ 0.
Hence, we find that S

n → 0 as n → ∞. Finally, invoking (3.34), we obtain S (2) n ≤ Ȟ -

1 2 k 1 n ⌊nv⌋ j=⌊ns⌋+3 k κ=1 j -1 n -s κ Ȟ-3 2 = Ȟ - 1 2 k ⌊nv⌋ n ⌊ns⌋+2 n k κ=1 ⌊ny⌋ + 1 n - 1 n -s κ Ȟ-3 2 dy ≤ Ȟ - 1 2 k ⌊nv⌋-1 n ⌊ns⌋+1 n k κ=1 (y -s κ ) Ȟ-3 2 dy ---→ n→∞ H - 1 2 k v s k κ=1 (y -s κ ) Ȟ-3 2 dy
and similarly by (3.35),

S (2) n ≥ Ȟ - 1 2 k 1 n ⌊nv⌋ j=⌊ns⌋+3 k κ=1 j n -s κ Ȟ-3 2 = Ȟ - 1 2 k ⌊nv⌋ n ⌊ns⌋+2 n k κ=1 ⌊ny⌋ + 1 n -s κ Ȟ-3 2 dy ≥ Ȟ - 1 2 k ⌊nv⌋+1 n ⌊ns⌋+3 n k κ=1 (y -s κ ) Ȟ-3 2 dy ---→ n→∞ Ȟ - 1 2 k v s k κ=1 (y -s κ ) Ȟ-3 2 dy.
(The convergence of the bounding integrals above, as n → ∞, is ensured by Lebesgue's dominated convergence theorem.) Thus, the convergence (3.33) follows from the sandwich lemma.

Functional convergence

To show that Theorems 2.11 and 2.18 indeed hold in the functional sense, we need to establish tightness of the relevant families of processes in the space D([0, 1] d ). To this end, we use the tightness criterion due to Bickel and Wichura [6, Theorem 3]. To apply this criterion, we need to bound the fourth moments of the increments of U

f uniformly over n ∈ N.

Moment bound and diagrams

As a preparation for the proof of tightness, we establish a moment bound for non-linear functionals of stationary Gaussian processes indexed by N d . The bound is a multiparameter extension of Proposition 4.2 of [START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence[END_REF], albeit under more restrictive assumptions. i ] = 1 for any i ∈ N d . Moreover, suppose that there exists a function ρ :

Z d → [-1, 1] such that E[Y i (1) Y i (2) ] = ρ(i (1) -i (2)
) for any i (1) , i (2) ∈ N d . If p ∈ {2, 3, . . .} and the Hermite coefficients a k , a k+1 , . . . of the function f satisfy

C ′′′′ (f, p) := ∞ k=k (p -1) k/2 √ k!|a k | < ∞, then for any l ∈ N d , E l -1/2 1≤i≤l f (Y i ) p ≤ 2 d C ′′′′ (f, p) 2 |i|<l |ρ(i)| k p/2 .
The proof of Proposition 4.2 of [START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence[END_REF] is based on a graph theoretic argument that involves multigraphs. We prove Lemma 4.1 using slightly different (but essentially analogous) formalism based on diagrams, defined below. Breuer and Major [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF] used diagrams to prove their central limit theorem for non-linear functionals of Gaussian random fields via the method of moments. In fact in the proof of Lemma 4.1, we adapt some of the arguments used in [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF]. (2) The degree of any vertex v ∈ V G is one.

(3) Any edge (j, k), (j ′ , k ′ ) ∈ E G has the property that j = j ′ .

We denote the class of diagrams of order (k 1 , . . . , k p ) by G(k 1 , . . . , k p ). For the sake of completeness we set G(k 1 , . . . , k p ) := ∅ when k 1 + • • • + k p is an odd number (no diagrams can then exist by the handshaking lemma of graph theory). Let us also define two functions λ 1 and λ 2 of an edge e = (j, k), (j ′ , k ′ ) ∈ E G , where j < j ′ , by setting λ 1 (e) := j and λ 2 (e) := j ′ . Diagrams are connected to Hermite polynomials and Gaussian random variables via the so-called diagram formula, which is originally due to Taqqu [START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence[END_REF]Lemma 3.2]. Below, we state a version of the formula that appears in [7, p. 431]. i ] = 1 for any i ∈ {1, . . . , p}. For any (k 1 , . . . , k p ) ∈ N p , we have

E p j=1 P k j (Y j ) = G∈G(k 1 ,...,kp) e∈E G E Y λ 1 (e) Y λ 2 (e) ,
where a sum over an empty index set is interpreted as zero.

Remark 4.4. The diagram formula can be used to estimate the cardinalities of classes of diagrams. As pointed out by Bardet and Surgailis [2, p. 461], using Lemma 4.3 and Lemma 3.1 of [START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence[END_REF] in the special case

Y := Y 1 = • • • = Y p , we obtain |G(k 1 , . . . , k p )| = E p j=1 P k j (Y ) ≤ (p -1) (k 1 +•••+kp)/2 k 1 ! • • • k p !. (4.5)
Proof of Lemma 4.1. Fix l ∈ N d . Let us define for any K ≥ k, a polynomial function

f K (x) = K k=k a k P k (x), x ∈ R.
By Fatou's lemma, Lemma 4.3, and inequality (4.5), it follows that

E f (Y i ) -f K (Y i ) p ≤ ∞ k 1 ,...,kp=K+1 |a k 1 • • • a kp ||G(k 1 , . . . , k p )| ≤ ∞ k=K+1 (p -1) k/2 √ k!|a k | p ----→ K→∞ 0
for any i ∈ N d . Thus, if ε > 0, then there exists K(l) ∈ N such that

E l -1/2 1≤i≤l f (Y i ) p -E l -1/2 1≤i≤l f K(l) (Y i ) p ≤ ε, (4.6)
by Minkowski's inequality and the fact that

X n L p (Ω) ----→ X implies E[X p n ] → E[X p ] when p ∈ {2, 3, . . .}. Lemma 4.3 yields now the expansion E l -1/2 1≤i≤l f K(l) (Y i ) p = l -p/2 1≤i (j) ≤l j∈{1,...,p} K(l) k 1 ,...,kp=k a k 1 • • • a kp G∈G(k 1 ,...,kp) e∈E G E Y i (λ 1 (e)) Y i (λ 2 (e)) = K(l) k 1 ,...,kp=k a k 1 • • • a kp G∈G(k 1 ,••• ,kp) I G (l),
where

I G (l) := l -p/2 1≤i (j) ≤l j∈{1,...,p} e∈E G ρ i (λ 1 (e)) -i (λ 2 (e)) , G ∈ G(k 1 , . . . , k p ). (4.7)
By Lemma 4.8, below, and inequality (4.5), we obtain the bound

K(l) k 1 ,...,kp=k a k 1 • • • a kp G∈G(k 1 ,••• ,kp) I G (l) ≤ K(l) k=k (p -1) k/2 √ k!|a k | p 2 d |i|<l |ρ(i)| k p/2 ≤ 2 d C ′′′′ (f, p) 2 |i|<l |ρ(i)| k p/2 .
In view of (4.6),

E l -1/2 1≤i≤l f (Y i ) p ≤ 2 d C ′′′′ (f, p) 2 |i|<l |ρ(i)| k p/2
+ ε, and letting ε → 0 concludes the proof.

The key ingredient in the proof of Lemma 4.1 was the following uniform bound for the absolute value of the quantity I G (l). We will derive this bound adapting the asymptotic analysis of the moments of a non-linear functional of a Gaussian random field, carried out in [7, pp. 435-436].

(p) ) such that (j, k), (j ′ , k ′ ) ∈ E G if and only if (σ -1 (j), k), (σ -1 (j ′ ), k ′ ) ∈ E G . For such a diagram G it holds that I G (l) = I G(l).
Relying on this invariance property we assume, without loss of generality, that

k 1 ≤ k 2 ≤ • • • ≤ k p-1 ≤ k p . (4.9)
Let us introduce the notation k G (j) := |{e ∈ E G : λ 1 (e) = j}| ∈ {0, 1, . . . , k j } for any j ∈ {1, . . . , p}. Since λ 1 (e) < λ 2 (e) for any e ∈ E G , we have

|I G (l)| ≤ l -p/2 1≤i (κ) ≤l κ∈{1,...,p} p j=1 e∈E G λ 1 (e)=j ρ i (j) -i (λ 2 (e))
= l -p/2 1≤i (κ) ≤l κ∈{2,...,p} p j=2 e∈E G λ 1 (e)=j ρ i (j) -i (λ 2 (e))

1≤i (1) ≤l e∈E G λ 1 (e)=1 ρ i (1) -i (λ 2 (e)) .

(4.10) Using Young's inequality (see [7, p. 435]) and the trivial estimate sup 1≤i≤l 1≤i (1) ≤l

ρ i (1) -i q ≤ |i|<l |ρ(i)| q , q ≥ 0,
one can show that sup 1≤i (κ) ≤l κ∈{2,...,p} 1≤i (1) 1) .

≤l e∈E G λ 1 (e)=1 ρ i (1) -i (λ 2 (e)) ≤ |i|<l |ρ(i)| k G (
Applying this procedure, mutatis mutandis, to (4.10) repeatedly we arrive at

|I G (l)| ≤ l -p/2 p j=1 |i|<l |ρ(i)| k G (j) . (4.11)
By Hölder's inequality, we have for any j ∈ {1, . . . , p},

|i|<l |ρ(i)| k G (j) ≤ 2l 1-k G (j)/k j |i|<l |ρ(i)| k j k G (j)/k j ≤ 2l 1-k G (j)/k j |j|<l |ρ(i)| k k G (j)/k j ,
where we use the proviso k j ≥ k to deduce the second inequality. Returning to (4.11), we have thus established that

|I G (l)| ≤ 2 d p/2 2l p/2-p j=1 k G (j)/k j |i|<l |ρ(i)| k p j=1 k G (j)/k j . ( 4
|ρ(i)| k p j=1 k G (j)/k j = |i|<l |ρ(i)| k p j=1 k G (j)/k j -p/2 |i|<l |ρ(i)| k p/2 ≤ 2l p j=1 k G (j)/k j -p/2 |i|<l |ρ(i)| k p/2 . ( 4

.14)

The assertion follows now by applying (4.14) to (4.12).

Remark 4.15. Strictly speaking, the inequality (4.13) is shown in [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF] as a part of a more extensive argument that uses the assumption that the diagram G is not regular (see [7, p. 432] for the definition of regularity). However, the assumption of non-regularity of G is completely immaterial concerning the validity of (4.13) and, in fact, not used in the proof in [7, p. 436].

Tightness

Furnished with the moment bound of Lemma 4.1, we prove the following lemma that enables us to complete the proofs of Theorems 2.11 and 2.18.

Lemma 4.16 (Tightness). Suppose that H ∈ (0, 1) d and that Assumption 2.9 holds. Then, the family U

(n) f : n ∈ N is tight in D([0, 1] d ).
Proof. The assertion follows from Theorem 3 of [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF], provided that sup

n∈N d sup s, t∈[0,1] d s<t E U (n) f ([s, t)) 4 t -s 2 < ∞. (4.17)
But since for any n ∈ N, the realization of U

f is constant on any set of the form

i -1 m(n) , i m(n) , 1 ≤ i ≤ m(n),
it suffices to show (see [6, p. 1665]) that

sup n∈N sup s, t∈En s<t E U (n) f ([s, t)) 4 t -s 2 < ∞, (4.18)
where

E n := {i/m(n) : 0 ≤ i ≤ m(n)},
E U (n) f ([s, t)) 4 t -s 2 = sup n∈N m(n) c(n) 2 sup 1≤l≤m(n) E l -1/2 1≤i≤l f X (n) i 4 ≤ sup n∈N 2 d C ′′′′ (f, 4) d ν=1 ψ ν (n) 2 ,
where

ψ ν (n) :=                    1 m ν (n) 1-2k(1-Hν ) |j|<mν (n) |r Hν (j)| k ≤ C ′ (H ν , k), H ν ∈ 1 -1 2k , 1 , 1 log m ν (n) |j|<mν (n) |r Hν (j)| k ≤ C ′ (H ν , k), H ν = 1 -1 2k , |j|<mν (n) |r Hν (j)| k ≤ j∈Z |r Hν (j)| k < ∞, H ν ∈ 0, 1 -1 2k .
(The first two inequalities above follow from the estimate (3.6).) We have, thus, verified the tightness condition (4.18).

Proof of Theorem 2.11. Recall that, for a family of pairs of random elements, tightness of marginals implies joint tightness. Thus, it follows from Lemma 4.16 that the family Z, U

f : n ∈ N is tight in D([0, 1] d ) 2 . (n)
The assertion follows then from Proposition 3.15 and Theorem 2 of [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF].

Proof of Theorem 2.18. Analogously to the proof of Theorem 2.11, above, we deduce from Lemma 4.16 that Λ

1 2 H,f Z, U (n) f : n ∈ N is tight in D([0, 1] d ) 2 . Moreover, Proposition 3.26 implies that U (n) f (t) P ---→ n→∞ Λ 1 2 H,f Z(t), t ∈ [0, 1] d ,
which, in turn, implies the corresponding convergence of finite-dimensional laws. Thus, by Theorem 2 of [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF], we have

Λ 1 2 H,f Z, U (n) f L ---→ n→∞ Λ 1 2 H,f Z, Λ 1 2 H,f Z in D([0, 1] d ) 2 . ( 4
U (n) f -Λ 1 2 H,f Z L ---→ n→∞ 0 in D([0, 1] d ). ( 4

.20)

It remains to note that the convergence (4.20) holds also in probability as the limit is deterministic.

5 Application to power variations

Convergence of power variations and their fluctuations

As an application of Theorems 2.11 and 2.18, we study the asymptotic behaviour of signed power variations of the fBs Z. Let p ∈ N be fixed throughout this section. We consider a family {V (n) p

: n ∈ N} of d-parameter processes, given by

V (n) p (t) := m(n) pH-1 1≤i≤⌊m(n)t⌋ Z i -1 m(n) , i m(n) p , t ∈ [0, 1] d , n ∈ N.
The realizations of V (n) p belong to the space D([0, 1] d ), as was the case with generalized variations. To describe the asymptotic behaviour of V

(n) p , we introduce v p (t) := γ p t , t ∈ [0, 1] d , ρ p (y) := y p -γ p , y ∈ R,
where γ p is the p-th moment of the standard Gaussian law, that is,

γ p := R y p γ(dy) =      0, p is odd, p/2 j=1 (2j -1), p is even.
Since the function ρ p is a polynomial, it belongs to L 2 (R, γ) and is a linear combination of finitely many Hermite polynomials. Moreover, it is easy to check that the Hermite rank of ρ p is given by k = 1, p is odd, 2, p is even.

Thus, the Hermite coefficients of ρ p satisfy Assumption 2.9. In what follows, we denote by Λ H,ρp the constant given by (2.13), substituting f with ρ p therein. As a straightforward application of Theorems 2.11 and 2.18, we obtain a functional law of large numbers (FLLN) for V (n) p as n → ∞. Proposition 5.1 (FLLN). We have

V (n) p P ---→ n→∞ v p in D([0, 1] d ).
Proof. For any t ∈ [0, 1] d and n ∈ N, the following decomposition holds

V (n) p (t) = c(n) 1 2
m(n) U

by Theorems 2.11 and 2.18 (and the non-randomness of the limit). Since the limits in (5.3) and (5.4) are continuous and addition is a continuous operation on C([0, 1] d ) 2 (with respect to the Skorohod topology), the assertion follows from the continuous mapping theorem.

Fluctuations of the process V (n) p around the FLLN limit v p , as n → ∞, are also of interest. The decomposition (5.2) suggests a non-trivial limit for the rescaled fluctuation process m(n) c(n)

1 2 V (n) p (t) -v p (t) , t ∈ [0, 1] d , (5.5)
In fact, we can write for any t ∈ [0, 1] d and n ∈ N,

m(n) c(n) 1 2 V (n) p (t) -v p (t) = U (n) ρp (t) -β (n) p (t), (5.6)
where

β (n) p (t) := m(n) c(n) 1 2 v p (t) -v p ⌊m(n)t⌋ m(n) ≥ 0.
If p is odd, then β (n) p = 0 = v p for any n ∈ N. Recalling that k = 1, Theorems 2.11 and 2.18 imply the following result, which extends Theorem 10 of [START_REF] Nualart | Central limit theorems for multiple stochastic integrals and Malliavin calculus[END_REF] (see also [START_REF] Burdzy | Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion[END_REF] for a related result).

Corollary 5.7 (Odd power variations). Suppose that p is odd.

(1) If H ∈ (0, 1) d \ 1 2 , 1 d , then

Z, m(n) c(n) 1 2 V (n) p L ---→ n→∞ Z, Λ H,ρp Z in D([0, 1] d ) 2 ,
where Z is the fBs of Theorem 2.11.

(

) If H ∈ 1 2 , 1 d , then m(n) c(n) 1 2 V (n) p P ---→ n→∞ a 1 Z in D([0, 1] d ), 2
(cf. Remark 2.20).

Suppose that p is even. In the special case d = 1, it is not difficult to see that β 2 → 0 as n → ∞ for any t ∈ [0, 1]. Thus, some of the existing functional limit theorems for power variations of fBm (e.g., Theorem 3 of [START_REF] Corcuera | Power variation of some integral fractional processes[END_REF] with an even power) can be recovered from Theorems 2.11 and 2.18 due to the decomposition (5.6). However, when d ≥ 2, the fluctuation process (5.5) may fail to converge in D([0, 1] d ). The reason for this is that, while β where ξ (n) (t) is some convex combination of n -1 ⌊nt⌋ and t. We will now show that β n cannot converge to a continuous function in D([0, 1] d ) as n → ∞ (similar, but slightly longer, argument shows that a discontinuous limit in D([0, 1] d ) is also impossible).

To this end, suppose that β The fluctuation process, analogous to (5.5), obtained by substituting the power variation V (n) p with its multilinear interpolation V (n) p := L n V (n) p satisfies the following functional limit theorem. In particular, it applies with any d ∈ N and p ∈ N.

Z, m(n) c(n) 1/2 V (n) p -v p L ---→ n→∞ Z, Λ H,ρp Z in C([0, 1] d ) 2 ,
where Z is the fBs of Theorem 2.11.

(

) If H ∈ 3 4 , 1 d , then m(n) c(n) 1/2 V (n) p -v p P ---→ n→∞ Λ H,ρp Z in C([0, 1] d ), 2
where Z is the Hermite sheet of Theorem 2.18.

The proof of Theorem 5.18 is based on the following two simple lemmas concerning the multilinear interpolation operators. First, we show that the function v p is a fixed point of the operator L n for any n ∈ N. ⌊m ν (n)t ν ⌋ + j m ν (n) {m ν (n)t ν } j (1 -{m ν (n)t ν }) 1-j .

It remains to observe that for any ν ∈ {1, . . . , d}, j∈{0,1}

⌊m ν (n)t ν ⌋ + j m ν (n) {m ν (n)t ν } j (1 -{m ν (n)t ν }) 1-j = ⌊m ν (n)t ν ⌋ + {m ν (n)t ν } m ν (n) = t ν ,
and the assertion follows.

Second, we show that convergence in probability in the space D([0, 1] d ) can be converted to convergence in probability in C([0, 1] d ) via interpolations.

  due to the relation (3.12) and Lemma 3.7. The expansion (2.7) and the connection of Hermite polynomials and multiple Wiener integrals (3.11) allows then us to write

k

  (t, •) n∈N , the convergence (3.31) holds also in H ⊗k . Clarke De la Cerda and Tudor [9, pp. 4-6] have shown

Lemma 4 . 1 (

 41 Moment bound). Let f be as in §2 and {Yi : i ∈ N d } a Gaussian process such that E[Y i ] = 0 and E[Y 2

Definition 4 . 2 .

 42 Let p ∈ {2, 3, . . .} and (k 1 , . . . , k p ) ∈ N p be such thatk 1 + • • • + k p is an even number. A diagram of order (k 1 , . . . , k p ) is a graph G = (V G , E G )with the following three properties:(1) We have V G = p j=1 {(j, 1), . . . , (j, k j )}.

Lemma 4 . 3 (

 43 Diagram formula). Let p ∈ {2, 3, . . .} and let Y 1 , . . . , Y p be jointly Gaussian random variables with E[Y i ] = 0 and E[Y 2

Lemma 4 . 8 . 2 ,

 482 For any k 1 , . . . , k p ≥ k, G ∈ G(k 1 , . . . , k p ), and l ∈ N d , |I G (l)| ≤ 2 d |i|<l |ρ(i)| k p/where I G (l) is defined by (4.7). Proof. As pointed out by Breuer and Major [7, p. 435], the quantity I G (l) is invariant under permutation of the levels of the diagram G. More precisely, if σ is a permutation of the set {1, . . . , p}, then we define a new diagram G ∈ G(k σ(1) , . . . , k σ

1 2

 1 n→∞ v p in D([0, 1] d ).(5.3)Note that c ν (n) /m ν (n) → 0 as n → ∞ for any ν ∈ {1, . . . , d}, whence D([0, 1] d ),

  (n) p (t) < c(n) -1

Example 5 . 8 . 4 d

 584 lattice E n = {i/m(n) : 0 ≤ i ≤ m(n)} for any n ∈ N, fluctuations of β (n) pin the complement of E n may be non-negligible or even explosive when n → ∞, as the following example shows. Consider the case where d ≥ 2, m(n) := (n, . . . , n) for any n ∈ N, and H ∈ 0,3 . Then we have by the mean value theorem,β (n) p (t) = n d/2-1 d ν=1 κ =ν ξ (n) κ (t) {nt ν }, t ∈ [0, 1] d , n ∈ N,

→ 9 )

 9 β in D([0, 1] d ), where β ∈ C([0, 1] d ). Then, it follows that β (n) p → β uniformly. By the continuity of β, there exists an open set E Note that there exists n 0 ∈ N such that E ∩ E n = ∅ for any n ≥ n 0 . Moreover, we can find n 1 ≥ n 0 such that inf 1 for any n ≥ n 1 . Thus, we find that for any n ≥ n 1 , sup t∈E β (n) p (t) ≥ n d/2-1 2 d-1 , (5.10)

Theorem 5 . 18 (

 518 Interpolated power variations). (1) If H ∈ (0, 1) d \ 3 4 , 1 d , then

Lemma 5 . 19 (

 519 Fixed point). We have L n v p = v p for any n ∈ N.Proof. Let t ∈ [0, 1] d and n ∈ N. By rearranging, we obtain that(L n v p )(t) = i∈{0,1} d γ p ⌊m(n)t⌋ + i m(n) {m(n)t} i (1 -{m(n)t})

Lemma 5 . 20 (

 520 Convergence and interpolation). Let X 1 , X 2 , . . . be random elements inD([0, 1] d ) and X a random element in C([0, 1] d ), all defined on a common probability space. If X n P -→ X in D([0, 1] d ) as n → ∞, then L n X n P ---→ n→∞ X in C([0, 1] d ).

Acknowledgements M. S. Pakkanen wishes to thank CEREMADE, Université Paris-Dauphine for warm hospitality and acknowledge support from CREATES (DNRF78), funded by the Danish National Research Foundation, from the Aarhus University Research Foundation regarding the project "Stochastic and Econometric Analysis of Commodity Markets", and from the Academy of Finland (project 258042).

→ β uniformly, the estimate (5.9) is not compatible with (5.10) and (5.11), which is a contradiction. (This also shows that β (n) p cannot converge to β along a subsequence.) Remark 5.12. Since β (n) p (t) = 0 if t ∈ E n , for any n ∈ N, choosing the sequence m(n) n∈N so that E 1 ⊂ E 2 ⊂ • • • (for example, set m(n) := 1 2 n , . . . , 1 2 n for any n ∈ N), one can ensure that the fluctuation process (5.5) convergences pointwise on a dense subset of [0, 1] d to a limit given by Theorem 2.11 or 2.18.

Multilinear interpolations

We have seen above that the rescaled fluctuations (5.5) of the power variations V (n) p , n ∈ N, around their LLN limit v p do not necessarily satisfy a functional limit theorem in D([0, 1] d ) when d ≥ 2 and p is even. Note that it is implicit in the definition of V (n) p that the corresponding partial sums are interpolated in a piecewise constant manner. Such an interpolation can have very poor precision in higher dimensions. In fact, interpolating V (n) p using a more appropriate, multilinear method enables functional convergence in the general case.

where the weights (2) The definition (5.14) involves slight abuse of notation. Namely,

when t ν = 1 and i ν = 1 for some ν ∈ {1, . . . , d}. But then α

(n) i (t) = 0, whence (5.17) is of no concern.

Proof. By (5.15), we can write for any t ∈ [0, 1] d and n ∈ N,

Thus, invoking (5.15) again, we obtain the bound sup

where

is the modulus of continuity of X, which satisfies lim u→0 w X (u) = 0 a.s. since the realizations of X are uniformly continuous. Thus, lim n→∞ w X m(n) -1 = 0 a.s. Finally, since convergence to a continuous function in

Proof of Theorem 5.18. We have for any n ∈ N, by Lemma 5.19 and the linearity of the operator L n ,

Note that the function t → v p ⌊m(n)t⌋ m(n) coincides with v p on E n . Since L n g depends on the function g only through the values of g on E n , we find that

The assertion in the case (2) follows now from Theorem 2.18 and Lemma 5.20. In the case (1) one can apply Theorem 2.11, Lemma 5.20, and Skorohod's representation theorem [START_REF] Kallenberg | Foundations of modern probability[END_REF]Theorem 4.30].

A Extension to Gaussian moving averages

We outline briefly, how Theorem 2.11 can be extended beyond the fBs setting. Namely, with minimal changes to the proof, we can also derive a central limit theorem for generalized variations of a multi-parameter Gaussian moving average process

as long as the kernel Ǧ(d) has the product form

where for any ν ∈ {1, . . . , d} we assume that Ǧ(1)

(A.1) Modifying the proof of Lemma 3.7 and following [10, p. 2557] one can show that the correlation structure of the increments (A.1) converges to the correlation structure of the corresponding increments of the fBs, that is,

Moreover, adapting the proof of Lemma 1 in [START_REF] Barndorff-Nielsen | Power variation for Gaussian processes with stationary increments[END_REF] one can construct a function ρ :

≤ ρ i (1) -i (2) . (A.3) Using (A.2), (A.3), and the Lebesgue's dominated convergence theorem, it is possible to modify the proof of Theorem 2.11 to establish that the processes Ž and 1 m(n)

where f is as in Theorem 2.11 with Hermite rank k, converge as n → ∞ in law to Ž, Λ

H,f Z in the space D([0, 1] d ) 2 , where Z is the fBs, or in this case a standard Brownian sheet, of Theorem 2.11.