Domenico Marinucci

Giovanni Peccati

Ergodicity and Gaussianity for Spherical Random Fields *

Keywords: Spherical Random Fields, High-Frequency Gaussianity, High-Frequency Ergodicity, Gaussian Subordination, Clebsch-Gordan Coefficients, Cosmic Variance, PACS: 02.50-r, 98.70-Vc, 98.80-k, AMS Classification: 60G60; 60G15, 60K40, 62M15, 42C10

We investigate the relationship between ergodicity and asymptotic Gaussianity of isotropic spherical random fields, in the high-resolution (or high-frequency) limit. In particular, our results suggest that under a wide variety of circumstances the two conditions are equivalent, i.e. the sample angular power spectrum may converge to the population value if and only if the underlying field is asymptotically Gaussian, in the high frequency sense. These findings may shed some light on the role of Cosmic Variance in Cosmic Microwave Background (CMB) radiation data analysis.

Introduction and background 1.Overview

The usual framework for proving asymptotic results in probability (for instance, central limit theorems or laws of large numbers) lies within the so-called large sample paradigm, according to which more and more (independent or weakly dependent) random variables are generated, and the limiting behaviour of some functionals of these variables (e.g., averages or empirical moments) is studied.

Physical applications, however, are prompting the development of a stochastic asymptotic theory of a rather different nature, where the indefinite repetition of a single experience is no longer available, and one relies instead on observations of the same (fixed) phenomenon with higher and higher degrees of resolution.

One crucial instance of this situation appears when dealing with the statistical analysis of random fields indexed by compact manifolds, the quintessential example being provided by the case of the sphere S 2 . Indeed, we are especially concerned with issues arising from the analysis of the Cosmic Microwave Background (CMB) radiation, a theme which is currently at the core of physical and cosmological research, see for instance [START_REF] Dodelson | Modern Cosmology[END_REF][START_REF] Kolb | The Early Universe[END_REF] for textbook references, and [START_REF] Spergel | First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameter[END_REF][START_REF] Spergel | Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Cosmology[END_REF][START_REF] Komatsu | Five-Year Wilkinson Microwave Anisotropy Probe Observations: Cosmological Interpretation[END_REF] for further discussions around the latest experimental data.

It is well-known that the CMB is a relic electromagnetic radiation providing a snapshot of the Universe at the so-called age of recombination, i.e. at the era when electrons in the primordial fluid arising from the Big Bang were captured by protons to form stable hydrogen atoms. Since the cross-section of hydrogen atoms is much smaller than for free electrons, after recombination photons can be viewed as diffusing freely across the Universe (to first order approximations). According to the latest experimental evidence, this has occurred some 3.7 × 10 5 years after the Big Bang, i.e. 13.7 billion years from the current epoch. Several experiments have been devoted to collecting extremely refined observations of the CMB, the leading role being played by the currently ongoing NASA mission WMAP (launched in 2001, see http://map.gsfc.nasa.gov/) and the ESA mission Planck, which is just now starting to operate after the launch on May 14, 2009 (see http://www.sciops.esa.int/).

From a mathematical point of view, the CMB can be regarded as a single realization of an isotropic, zero-mean, finite variance spherical random field, for which the following spectral representation holds (see e.g. [START_REF] Adler | Random Fields and Geometry[END_REF] or [START_REF] Leonenko | Limit Theorems for Random Fields with Singular Spectrum[END_REF])

T (x) = ∞ l=0 l m=-l a lm Y lm (x) , x ∈ S 2 .
(1)

Here, the collection {Y lm : l ≥ 0, m = -l, ..., l} stands for the usual triangular array of spherical harmonics, which are well-known to provide a complete orthonormal system for the L 2 (S 2 ) space of square-integrable functions (with respect to Lebesgue measure) on the sphere -see [START_REF] Stein | Introduction to Fourier analysis on Euclidean spaces[END_REF][START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF][START_REF] Vilenkin | Representation of Lie Groups and Special Functions[END_REF]. In a loose sense, we can say that the frequency parameter l is related to a characteristic angular scale, say ϑ l , according to the relationship ϑ l ≃ π/l. The (random) triangular array of spherical harmonic coefficients {a lm : l ≥ 0, m = -l, ..., l} are such that Ea lm = 0 and Ea lm a l ′ m ′ = C l δ l ′ l δ m ′ m , the bar denoting complex-conjugation and δ b a indicating the Kronecker delta function. The non-negative sequence {C l : l ≥ 0} (not depending on m -see [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF] as well as the forthcoming section) is the angular power spectrum of the spherical field (see for instance [START_REF] Baldi | Some characterizations of the spherical harmonics coefficients for isotropic random fields[END_REF][START_REF] Baldi | On the characterization of isotropic random fields on homogeneous spaces of compact groups[END_REF]).

As recalled above, our work deals with asymptotic issues, where the expression "asymptotic" has to be understood in the high-resolution (or high-frequency) sense. This means that we focus on the behaviour of the Fourier components

T l (x) := l m=-l a lm Y lm (x) , x ∈ S 2 , l ≥ 0, (2)
associated with a fixed spherical field, as the frequency l grows larger and larger (plainly, each T l is the projection of the field T into the orthogonal subspace of L 2 S 2 spanned by the spherical harmonics {Y lm : m = -l, ..., l}). Note that this is the typical framework faced by experimentalists handling satellite missions as those mentioned above. Indeed, these missions are observing the same (unique) realization of our Universe on the so-called last scattering surface; more recent and more sophisticated experiments are then characterized by higher and higher frequencies (smaller and smaller scales) being observed. For instance, for the pioneering CMB mission COBE in 1989-1992 (which led to the Nobel Prize for Smoot and Mather in 2006) only frequencies in the order of a few dozens were recorded (i.e., scales of several degrees), a limit which was raised to few hundreds by WMAP (i.e., approximately a quarter of degree) and is expected to grow to a few thousands with Planck (i.e., a few arcminutes).

The principal goal of this paper is to enlighten some partial new connections between two high-resolution characterizations of spherical fields, that is, ergodicity and asymptotic Gaussianity. Roughly speaking (formal details are given in the forthcoming Sections 1.2 and 1.3), one says that the spherical field T is ergodic if the empirical version of the power spectrum of T (see formula (3) below) can be used as a consistent estimator of the sequence {C l } (at least for high values of l). On the other hand, we say that T is asymptotically Gaussian, whenever suitably normalized versions of the frequency components of T l exhibit Gaussian fluctuations for high values of l. As discussed below, these two notions are tightly connected whenever one deals with fields having an isotropic (or, equivalently, rotationally-invariant) law.

Remark. For the rest of the paper, every random object is defined on a suitable (common) probability space (Ω, F, P ) .

High-frequency ergodicity

In what follows, we shall consider a real-valued random field T = T (x) : x ∈ S 2 indexed by the sphere S 2 . The random field T satisfies the following basic assumptions: (i) the law of T is isotropic, that is, T has the same law as x → T (gx) for every rotation g ∈ SO (3) (here, we select the canonical action of SO (3) on S 2 ); (ii) T is square-integrable and centered. Under assumptions (i)-(ii), the harmonic expansion (1) takes place, both in L 2 (P ) (for fixed x) and in the product space L 2 Ω × S 2 , P ⊗ dλ , where λ stands for the Lebesgue measure. Note that the last claim hinges on the fact that one can regard T as an application of the type T : Ω × S 2 → R : (ω, x) → T (ω, x). As anticipated in the previous section, another useful property of T (easily deduced from isotropy -see e.g. [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF]) is that the harmonic coefficients a lm are such that the power spectrum associated with T , defined as the collection {C l : l = 0, 1, ...} (with C l = E |a lm | 2 ), depends uniquely on the frequency indices l.

In physical experiments (for instance, when measuring the CMB radiation), the power spectrum of a given spherical field is usually unknown. For this reason, a key role is played by its empirical counterpart (called the empirical power spectrum -see for instance [START_REF] Efstathiou | Myths and truths concerning estimation of power spectra: the case for a hybrid estimator[END_REF], [START_REF] Polenta | Unbiased Estimation of Angular Power Spectra[END_REF]), which is given by

C l = 1 2l + 1 l m=-l |a lm | 2 , l = 0, 1, 2, ... (3)
An important issue to be addressed is therefore to establish conditions under which the distance between the quantities C l and C l converges to zero (in a sense that is defined below) when l → ∞, that is, when higher and higher frequencies of the expansion (1) are available to the observer. Although the asymptotic behaviour of spectrum estimators has been very deeply investigated for stochastic processes in Euclidean domains and under large sample asymptotics (see for instance [START_REF] Brockwell | Time series: theory and methods[END_REF][START_REF] Leonenko | Limit Theorems for Random Fields with Singular Spectrum[END_REF][START_REF] Yadrenko | Spectral theory of random fields[END_REF]), only basic results are known in the high-resolution setting.

For instance, it is immediate that the finite variance of T entails that, for every x ∈ S 2 ,

ET (x) 2 = l≥0 (2l + 1) 4π C l < ∞,
from which one deduces that C l → 0 and also

l≥0 E C l = l≥0 C l < ∞.
By reasoning as in the proof of the Borel-Cantelli Lemma we therefore infer that, for any ε > 0,

P lim l→∞ sup C l > ε ≤ lim l→∞ ℓ≥l P C ℓ ≥ ε ≤ lim l→∞ 1 ε ℓ≥l C ℓ = 0, (4)
yielding in turn that both C l and | C l -C l | almost surely converge to zero as l → ∞. Plainly, since this result does not provide any information about the magnitude of the ratio

| C l -C l |/C l ,
it is virtually useless for statistical applications. In particular, one cannot conclude from (4) that the estimation of C l based on C l is consistent in a satisfactory statistical sense.

Starting from these considerations, one sees that it is indeed necessary to focus on normalized quantities, such as the sequence

C l = 1 2l + 1 l m=-l |a lm | 2 C l = C l C l , l ≥ 0. (5)
Note that E C l = 1, and also that the coefficient C l is not observable (whereas C l is). The sequence C l : l ≥ 0 can be used in order to meaningfully evaluate the asymptotic performance of any statistical procedure based on C l . The following definition uses the coefficients C l in order to define ergodicity.

Definition 1 (HFE) Let T be an isotropic, finite variance spherical random field with angular power spectrum {C l : l ≥ 0} . We shall say that T is High-Frequency Ergodic (HFE -or ergodic in the high-frequency sense) if and only if

lim l→∞ E C l -1 2 = lim l→∞ E C l C l -1 2 = 0. (6)
Condition ( 6) implies of course that C l = C l /C l converges in probability towards the constant 1.

Remark. In some sense, the term "high-frequency consistency" seems to better describe property [START_REF] Brockwell | Time series: theory and methods[END_REF]. However, in the statistical literature consistency is usually viewed as a property of a sequence of estimators, whereas here we deal with a property of the field T , so that we find the term ergodicity more suitable. Another way of formulating this point is to say that (6) characterizes the ergodicity of the "normalized empirical spectral measure" C l : l ≥ 0 , as l diverges.

Ergodicity of Gaussian fields (and associated Gaussian fluctuations)

As an illustration (and for future reference) we now test Definition 1 under the additional assumption that T is Gaussian. In this case, it is readily seen that, for every l ≥ 1, the components of the vector {a lm : m = 1, ..., l} are complex-valued and independent. Moreover, the random quantities a l0 / √ C l , √ 2Re (a lm )/ √ C l and √ 2Im (a lm )/ √ C l (m = 0, ..., l) are independent and identically distributed N (0, 1) random variables (these facts are well-known, see e.g. [START_REF] Baldi | Some characterizations of the spherical harmonics coefficients for isotropic random fields[END_REF], [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF] and the references therein). It is now easy to prove that

C l = 1 2l + 1 l m=-l |a lm | 2 C l → 1, (7)
in every norm L p , p ≥ 1. Indeed, since a 2 l0 /C l law ∼ χ 2 1 and the set 2a 2 lm /C l : m = 1, ..., l is composed of i.i.d. χ 2 2 random variables independent of a l0 (here, χ 2 n denotes a standard chisquare distribution with n degrees of freedom),

E C l -1 2 = 1 (2l + 1) 2 E a 2 l0 C l -1 + 2 l m=1 |a lm | 2 C l -1 2 = 2 2l + 1 -→ l→∞ 0,
and one can use the fact that, for polynomial functionals of a Gaussian field of fixed degree, all L p topologies coincide.

We shall now provide (see the forthcoming Proposition 2) a CLT that is naturally associated with the convergence described in [START_REF] Cabella | Statistical challenges in the analysis of Cosmic Microwave Background radiation[END_REF]. Note that, instead of using the classic Berry-Esseen results (see e.g. Feller [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]), we rather apply some recent estimates (proved in [START_REF] Nourdin | Stein method on Wiener chaos[END_REF] and [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of the Gaussian Wiener chaos[END_REF] by means of infinite-dimensional Gaussian analysis and the so-called "Stein's method" for probabilistic approximations) allowing to compare, for fixed l, the total variation distance between the law of the normalized random variable 2l + 1 2

C l C l -1 = 2l + 1 2 C l -1 ,
and that of a standard Gaussian random variable. Recall that the total variation distance between the laws of two real-valued random variables X and Y is given by

d T V (X, Y ) = sup A |P (X ∈ A) -P (Y ∈ A)| ,
where the supremum runs over all Borel sets A.

Proposition 2 Let N (0, 1) denote a centered standard Gaussian random variable. Then, for all l ≥ 0 we have

d T V 2l + 1 2 C l C l -1 , N (0, 1) ≤ 8 2l + 1 , ( 8
)
so that, in particular, as l → ∞,

2l + 1 2 C l -1 law → N (0, 1) . (9)
Proof. We have 2l + 1 2

C l C l -1 = 1 2(2l + 1) a 2 l0 C l + l m=1 2 {Re a lm } 2 + {Im a lm } 2 C l -(2l + 1) = 1 (2l + 1) 2l+1 m=1 (x 2 lm -1) √ 2 ,
where {x lm } are a triangular array of i.i.d. standard Gaussian random variables. Standard calculations yield that

cum 4 √ 2 (2l + 1) 2l+1 m=1 (x 2 lm -1) 2 = 12 2l + 1 ,
where cum j stands for the jth cumulant. Now recall that in [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of the Gaussian Wiener chaos[END_REF] it is proved that, for every zero mean and unit variance random variable F q that belongs to the qth Wiener chaos associated with some Gaussian field (q ≥ 2), the following inequality holds:

d T V (F q , N (0, 1)) ≤ 2 q -1 3q cum 4 (F q ) .
The result now follows immediately, since each variable 2l+1 2 b C l C l -1 has unit variance, and is precisely an element of the second Wiener chaos associated with T.

It is simple to verify numerically that the convergence (9) takes place rather fast. For instance, for l = 100, the bound in total variation is of the order of 2%, while for l = 1000 we deduce an order of 0.6%.

We stress that the previous results heavily rely on the Gaussian assumption, and cannot be easily extended to the framework of non-Gaussian and isotropic spherical fields. The main reason supporting this claim is contained in the references [START_REF] Baldi | Some characterizations of the spherical harmonics coefficients for isotropic random fields[END_REF][START_REF] Baldi | On the characterization of isotropic random fields on homogeneous spaces of compact groups[END_REF], where it is shown that, under isotropy, the coefficients a lm are independent if and only if the underlying field is Gaussian, and this despite the fact that they are always uncorrelated by construction. In other words, sampling independent, non-Gaussian random coefficients to generate maps according to (1) will always yield an anisotropic random field. The dependence structure among the coefficients {a lm } is in general quite complicated, albeit it can be neatly characterized in terms of the group representation properties of SO(3) (see [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF] and [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF]). In view of this, to derive any asymptotic result for C l under non-Gaussianity assumptions for T , is by no means trivial and still almost completely open for research.

High-frequency Gaussianity

A different form of asymptotic theory has been addressed in an apparently unrelated stream of research, for instance in [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF].

Definition 3 (HFG) Let T (x) be an isotropic, finite variance spherical random field, and recall the notation ( 1) and [START_REF] Baldi | Some characterizations of the spherical harmonics coefficients for isotropic random fields[END_REF]. We say that T (x) is high-frequency Gaussian (HFG) whenever

T l (x) V ar {T l (x)} law → N (0, 1) , as l → ∞, (10)
for every fixed x ∈ S 2 .

Remark. It is more delicate to define HFG involving convergence in the sense of finite dimensional distributions. Indeed, in [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF] it is shown that, even if relation [START_REF] Diaconis | A dozen de Finetti-style results in search of a theory[END_REF] holds, the finitedimensional distributions of order ≥ 2 of the field x → T l (x)/ V ar {T l (x)} may not converge to any limit.

It is clear that a Gaussian field is asymptotically Gaussian: however, as shown in [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF], characterizing non-Gaussian fields that are HFG can be a difficult task, even if the underlying field T is a simple transformation (for instance, the square) of some Gaussian random function. Conditions for the HFG property to hold in some non-Gaussian circumstances are given in [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF], by using group representations -yielding some interesting connection with random walks on hypergroups associated with the power spectrum of T . We stress that the possible existence of HFG behaviour entails deep consequences on CMB data analysis. On one hand, in fact, parameter estimation on CMB data is largely dominated by likelihood approaches, whence an asymptotically Gaussian behaviour would great simplify the implementation of optimal procedures. On the other hand, testing for non-Gaussianity is a key ingredient in the validation of the so-called inflationary scenarios, and the possible existence of high frequency Gaussian components for non-Gaussian models might set a theoretical limit to the investigation in this area.

Purpose and plan

Our purpose in this paper is to investigate the relationships between the HFG and HFE properties under an assumption of Gaussian subordination, that is, by considering fields T that can be written as a deterministic function of some isotropic, real-valued Gaussian field. We will mainly focus on the case of polynomial subordinations, where the polynomials are of the Hermite type. Note also that Gaussian subordination is the favoured framework for CMB modeling in a non-Gaussian setting (see e.g. [START_REF] Bartolo | Non-Gaussianity from inflation: theory and observations[END_REF], [START_REF] Hu | The angular trispectrum of the CMB[END_REF], [START_REF] Yadav | Evidence of Primordial Non-Gaussianity (fNL) in the Wilkinson Microwave Anisotropy Probe 3-Year Data at 2.8σ[END_REF]).

Our main finding is that, despite their apparent independence, the HFG and HFE properties will turn out to be very close in a broad class of circumstances, suggesting that ergodicity (and hence the possibility to draw asymptotically justifiable statistical inferences) and asymptotic Gaussianity are very tightly related in a high-resolution setting. This may lead, we believe, to important characterizations of Gaussian random fields, and to a better understanding of the conditions for the validity of statistical inference procedures based on observations drawn from a unique realizations of a compactly supported random field, as in the spherical case.

The plan of this paper is as follows: in Section 2 we state and prove our main result, establishing necessary and sufficient conditions for ergodicity and Gaussianity and exploring the link between them. Indeed, these conditions turn out to be extremely close, so that in Section 3 we can indeed discuss more thoroughly a special case of practical relevance, namely the quadratic case. Section 4 is devoted to further discussion and directions for further research.

A general statement about Gaussian subordinated fields

The two notations (1) and ( 2) are adopted throughout the sequel. Let us first recall a few basic facts and definitions.

(I) The first point concerns a characterization of isotropy in terms of angular power spectra. Indeed, as discussed in [START_REF] Hu | The angular trispectrum of the CMB[END_REF][START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF], if a random field is isotropic with finite fourth-order moment, then there exists necessarily an array T l 3 l 4 l 1 l 2 (L) such that

cum {a l 1 m 1 , a l 2 m 2 , a l 3 m 3 , a l 4 m 4 } (11) = LM (-1) M l 1 l 2 L m 1 m 2 M l 3 l 4 L m 3 m 4 -M (2L + 1)T l 3 l 4 l 1 l 2 (L) .
In general, the symbol cum {X 1 , ..., X m } denotes the joint cumulant of the random variables X 1 , ..., X m . Also, we label as usual T l 3 l 4 l 1 l 2 (L) the cumulant trispectrum of the random field (see for instance [START_REF] Hu | The angular trispectrum of the CMB[END_REF], [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF]); as made clear by our notation, the quantity T l 3 l 4 l 1 l 2 (L) does not depend on m 1 , m 2 , m 3 , m 4 (this phenomenon is analogous to the fact that the power spectrum only depends on the frequency l -see [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF] for a discussion of this point). On the right-hand side of ( 11), we have also introduced the well-known Wigner's coefficients, which arise in the group representation theory of SO(3) and are discussed at length in many excellent monographs on the quantum theory of angular momentum -see e.g. [START_REF] Liboff | Introductory Quantum Mechanics[END_REF][START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF][START_REF] Vilenkin | Representation of Lie Groups and Special Functions[END_REF]. For future reference, we also recall that the Wigner's 3j coefficients are equivalent, up to normalization and phase factor, to the Clebsch-Gordan coefficients given by (see e.g. [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF])

C LM l 1 m 1 l 2 m 2 = (-1) l 1 -l 2 +m 3 √ 2L + 1 l 1 l 2 L m 1 m 2 M . ( 12
)
As noted by [START_REF] Hu | The angular trispectrum of the CMB[END_REF], geometrically the multipoles (l 1 , l 2 , l 3 , l 4 ) can be viewed as the sides of a quadrilateral, and L as one of its main diagonals; L is also the shared size of the two triangle formed by the corresponding pairs of side. Clebsch-Gordan coefficients ensure that the triangle conditions are satisfied, indeed they are different from zero only if. l 1 ≤ l 2 + L, l 2 ≤ l 1 + L, and

L ≤ l 1 + l 2 .
(II) We shall sometimes label a point x of the sphere S 2 in terms of its spherical coordinates, that is, x = (ϑ, ϕ), where 0 ≤ ϑ ≤ π and 0 ≤ ϕ < 2π.

(III) Easy considerations yield the important fact that, for any isotropic random field T ,

T l (ϑ, ϕ) law = T l (N ) = lm a lm Y lm (N ) law = a l0 2l + 1 4π ,
where we denote by N := (0, 0) the North Pole of the sphere and by " law = " the equality in law between two random elements.

(IV) It is immediate that, if T is isotropic, then for every deterministic function F the subordinated random application x → F (T (x)) is also isotropic. Moreover, if F (T (x)) is square integrable, then F (T (•)) also admits a harmonic expansion analogous to [START_REF] Adler | Random Fields and Geometry[END_REF]. One specific instance of this situation is obtained by choosing T to be Gaussian and isotropic, and F to be any of the Hermite polynomials {H q : q ≥ 0} (in this case, one talks about a Gaussian subordination of the Hermite type). We recall that the polynomials H q are such that H q = δ q 1, where 1 stands for the function which is constantly equal to one, δ 0 is the identity, and δ q (q ≥ 1) represents the qth iteration of the divergence operator δ, acting on smooth functions as δf (x) = xf (x)-f ′ (x) .

For instance H 0 = 1, H 1 (x) = x, H 2 (x) = x 2 -1, and so on. When T is Gaussian, we adopt the notation

H q (T (x)) := T q (x) = ∞ l=0 T l;q (x), x ∈ S 2 , q ≥ 2 , ( 13
)
where

T l;q (x) = l m=-l a lm;q Y lm (x) (14)
is the lth frequency component of T q , with a lm;q the associated harmonic coefficients. We shall also write {C l;q : l ≥ 0} and T ll ll (L; q) , respectively, for the power spectrum and for the cumulant trispectrum of T q , as introduced at Point (I). According to [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF]Theorem 3], one has that C l;q admits the following expansion in terms of the power spectrum {C l } of T :

C l;q = q! ∞ l 1 ,...,lq=0 C l 1 •••C lq 4π 2l + 1 q i=1 2l i + 1 4π L 1 ...L q-2 C L 1 ,L 2 ,...,L q-2 ,l;0 l 1 ,0;...;lq0 2 , ( 15
)
where C

L 1 ,L 2 ,...,L q-2 ,l;0 l 1 ,0;...;lq0

indicates a convolution of Clebsch-Gordan coefficients, that is

C λ 1 ,λ 2 ,...,λ p-1 ;µ l 1 ,m 1 ;...;lpmp := λ 1 µ 1 =-λ 1 ... λ p-2 µ p-2 =-λ p-2 C λ 1 ,µ 1 l 1 ,m 1 ,l 2 ,m 2 C λ 2 ,µ 2 λ 1 ,µ 1 ;l 3 ,m 3 • • • C λ p-1 ,µ λ p-2 ,µ p-2 ;lp,mp (16)
(see [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF] and [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF] for more details on these convolutions, which can also be viewed as probability amplitudes in alternative coupling schemes for quantum angular momenta, compare [START_REF] Biedenharn | The Racah-Wigner algebra in quantum theory[END_REF]).

(V) An easy but important remark is the following. Since the expansion (1) is in order, the law of a centered isotropic Gaussian field T is completely encoded by the power spectrum {C l : l ≥ 0}. This is a consequence of the fact that, in this case, the array {a lm : l ≥ 0, m = 0, ..., l} is composed of independent Gaussian random variables such that: (i) a l0 is real-valued, and (ii) for every m ≥ 1, the coefficient a lm has independent and equidistributed real and imaginary parts.

As anticipated, we shall now prove some new connections between HFE and HFG spherical fields (see Definitions 1 and 3), in the special case of fields of the type T q , as defined in [START_REF] Durren | Cosmology[END_REF]. In particular, our main finding (as stated in Theorem 4) Note that the conditions appearing in the following statement involve the coefficients C l;q given in [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], and that these coefficients are completely determined by the power spectrum of the underlying Gaussian field T .

Theorem 4 Let q ≥ 2, and define T q according to ( 13), where T is Gaussian and isotropic. Let T l 3 l 4 l 1 l 2 (L; q) be the reduced trispectrum of T . Introduce the notation Then, the following holds.

w 1l (L) := C L0
1. the random field T q is high-frequency Gaussian if and only if

lim l→∞ 2l L=0 w 1l (L) T ll ll (L; q) C 2 l;q = 0. ( 17
)
2. On the other hand, T q is high-frequency ergodic if and only if

lim l→∞ 2l L=0 w 2l (L) T ll ll (L; q) C 2 l;q = 0. ( 18
)
Before proving Theorem 4, we shall note that C L0 l0l0 2 is different from zero only for L even, and T ll ll (L) is not in general positive-valued. Moreover, in view of the forthcoming Lemma 5, also in [START_REF] Hu | Renormalized self-intersection local time for fractional Brownian motion[END_REF] the sum runs only over even values of L.

Lemma 5 T ll ll (L) is zero when L is odd

Proof. From [17, Eq. ( 17)], we infer that, in general,

T l 3 l 4 l 1 l 2 (L) = (-1) l 1 +l 2 +L T l 3 l 4 l 1 l 2 (L) .
Considering the case l 1 = l 2 = l 3 = l 4 = l, we obtain the desired result.

Proof of Theorem 4. (Proof of 1.) Consider the random spherical field (ϑ, ϕ) → Tl;q (ϑ, ϕ) := T l;q (ϑ, ϕ)

V ar T l;q (N ) , (ϑ, ϕ) ∈ [0, π] × [0, 2π),
where N is the North Pole, and observe that, by isotropy and for every (ϑ, ϕ), Tl;q (ϑ, ϕ) law = a l0 4πC l;q .

The field Tl;q is mean-zero and has unit variance: since it also belong to the qth Wiener chaos associated with T , we can deduce from the results in [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF] it is asymptotically Gaussian if and only if

lim l→∞ 1 C 2
l;q cum 4 {a l0;q } = 0 .

As discussed e.g. in [START_REF] Hu | The angular trispectrum of the CMB[END_REF] and [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF], isotropy entails that we can write the fourth-order cumulant as

cum 4 {a l0;q } = LM (-1) M l l L 0 0 M l l L 0 0 -M (2L + 1)T ll ll (L) = L l l L 0 0 0 2 (2L + 1)T ll ll (L) ,
so that the field is asymptotically Gaussian if and only if

lim l→∞ 1 C 2 l;q L l l L 0 0 0 2 (2L + 1)T ll ll (L) = 0 . ( 19
)
Since relation [START_REF] Dodelson | Modern Cosmology[END_REF] is in order, we write

l l L 0 0 0 2 (2L + 1) = C L0 l0l0 2 ,
entailing in turn that

L l l L 0 0 0 2 (2L + 1) = 2l L=0 C L0 l0l0 2 = 2l L=0 L M =-L C LM l0l0 2 ≡ 1 ,
where the second equality follows from the fact that Clebsch-Gordan coefficients C l 3 m 3 l 1 m 1 l 2 m 2 are different from zero only for m 3 = m 1 + m 2 , and the third equality is a consequence from the orthonormality properties of the coefficients (which are the elements of unitary matrices whose rows are indexed by m 1 , m 2 and whose columns are indexed by l 3 , m 3 ). We therefore have

1 C 2 l;q cum 4 {a l0;q } = 1 C 2 l;q L C L0 l0l0 2 T ll ll (L) ,
yielding the desired conclusion. (Proof of 2.) On the other hand, we obtain also

E C l;q C l;q -1 2 = V ar C l;q C l;q -1 = 1 (2l + 1) 2 1 C 2
l;q m 1 m 2 cum {a lm 1 ;q , a lm 1 ;q, a lm 2 ;q , a lm 2 ;q } + 2 (2l + 1) 1) m 1 +m 2 cum {a lm 1 ;q , a l,-m 1 ;q, a lm 2 ;q , a l,-m 2 ;q } + 2 (2l + 1)

2 1 C 2 l;q m E |a lm;q | 2 2 (20) = 1 (2l + 1) 2 1 C 2 l;q m 1 m 2 (-
= 2 (2l + 1) 2 1 C 2 l;q m 1 m 2 LM (-1) M +m 1 +m 2 l l L m 1 m 2 M × × l l L -m 1 -m 2 -M (2L + 1)T ll ll (L) + 2 (2l + 1) (22) = 2 (2l + 1) 2 1 C 2 l;q 2l L=0 L even (2L + 1)T ll ll (L) + 2 (2l + 1) . ( (21)
) 23
It is simple to notice that 2l L=0 (2L + 1) = 2 2l(2l + 1) 2 + 2l + 1 = (2l + 1) 2 , so that we have

E C l;q C l;q -1 2 = 2 2l L=0 L even w ll lL T ll ll (L) + 2 (2l + 1)
, where w lL ≥ 0 and

2l L=0 L even w lL = 1.
The result now follows immediately.

Remark. Note that

C L0 l0l0 2 = (2L + 1)( 2l+L 2 !) 2 ( L 2 !) 2 (L!) 2 (2l -L)! (2l + L + 1)! ≤ 1 (2L + 1) w 2l (L) = (2L + 1) (2l + 1) 2 ≤ 1 2l + 1
Note also that in the Gaussian case (e.g., q = 1) we have T ll ll (L) ≡ 0, whence

E C l;q C l;q -1 2 = 2 (2l + 1)
→ 0 , as expected.

The previous result strongly suggests that the conditions for asymptotic Gaussianity (HFG) and for ergodicity (HFE) should be tightly related. Indeed we conjecture that HFE and HFG are equivalent in the case of Hermite type Gaussian subordinations (and most probably even in more general circumstances). However, proving this claim seems analytically too demanding at this stage, so that for the rest of the paper we content ourselves with a detailed analysis of quadratic Gaussian subordinations. In particular, we believe that the content of the forthcoming Section 12 (which is already quite technical) may provide the seed for a complete understanding of the HFG-HFE connection.

Remark 6 It should be noted that the reduced trispectrum satisfies (see [17, Eq. ( 16)])

T ll ll (L ′ ) = L (2L + 1) l l L l l L ′ T ll ll (L) .
In the previous remark, we introduced the well-known Wigner's 6j coefficients, which intertwine alternative coupling schemes of three quantum angular momenta (see [START_REF] Biedenharn | The Racah-Wigner algebra in quantum theory[END_REF], [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF] for futher properties and much more discussion). Their relationship with Wigner's 3j coefficients is provided by the identity

a b e c d f := α,β,γ ε,δ,φ (-1) e+f +ε+φ a b e α β ε c d e γ δ -ε a d f α δ -φ c b f γ β φ (24
) (see [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF], Chapter 9, for analytic expressions and a full set of properties).

3 The quadratic case

The class D and main results

As anticipated, the purpose of this section is to provide a more detailed and explicit analysis of the quadratic case q = 2. For simplicity, in the sequel we consider a centered Gaussian isotropic spherical field T such that V ar(T (x)) = l (2l + 1)C l /4π = 1, where {C l } is as before the power spectrum of T. We start by recalling the notation

T 2 (x) = H 2 (T (x)) = ∞ l 1 ,l 2 =1 m 1 m 2 a l 1 m 1 a l 2 m 2 Y l 1 m 1 (x)Y l 2 m 2 (x) -1 , ( 25
)
where T is isotropic, centered and Gaussian. Our first result can be seen as a consequence of formula ( 15) (or, more generally, of the results of [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF]). Here, we provide a proof for the sake of completeness.

Lemma 7

The angular power spectrum of the squared random field ( 25) is given by

C l;2 = E|a lm;2 | 2 = 2 l 1 l 2 C l 1 C l 2 l 1 l 2 l 0 0 0 2 (2l + 1)(2l + 1)
4π .

Proof. Recall first that Y 00 (x) ≡ (4π) -1/2 , see [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF], equation 5.13.1.1. Hence, in view of ( 25), we have that, for l = 0,

a 00;2 = S 2    l 1 l 2 m 1 m 2 a l 1 m 1 a l 2 m 2 Y l 1 m 1 (x)Y l 2 m 2 (x) -1    Y 00 (x)dx = 1 √ 4π l 1 l 2 m 1 m 2 a l 1 m 1 a l 2 m 2 S 2 Y l 1 m 1 (x)Y l 2 m 2 (x)dx - √ 4π = 1 √ 4π l 1 l 2 m 1 m 2 a l 1 m 1 a l 2 m 2 δ l 2 l 1 δ m 2 m 1 - √ 4π = 1 √ 4π lm |a lm | 2 - √ 4π .
It follows that

Ea 00;2 = lm 2l + 1 √ 4π C l - √ 4π = √ 4π lm 2l + 1 4π C l -1 = 0 ,
and

EH 2 (T (x)) = E ∞ l=0
a lm;2 Y lm (x) = Ea 00;2 Y 00 (x) = 0 , the second step following because Ea lm = 0 for all l > 0 under isotropy (see [START_REF] Baldi | Some characterizations of the spherical harmonics coefficients for isotropic random fields[END_REF]). Indeed we have (from [START_REF] Marinucci | A central limit theorem and higher order results for the angular bispectrum[END_REF], and in view of ( 14))

a lm;2 = S 2 l 1 l 2 m 1 m 2 a l 1 m 1 a l 2 m 2 Y lm 1 (x)Y lm 2 (x)Y lm (x)dx = ∞ l 1 ,l 2 =1 m 1 m 2 a l 1 m 1 a l 2 m 2 l 1 l 2 l m 1 m 2 -m l 1 l 2 l 0 0 0 (2l 1 + 1)(2l 2 + 1)(2l + 1) 4π . ( 26
)
Note that the constant term -1 has no effect for l ≥ 1, because

S 2
Y lm (x)dx = 0 for all l ≥ 1 .

Now

Ea lm;2 = l 1 m 1 C l 1 (-1) m 1 l 1 l 2 l m 1 -m 1 -m l 1 l 2 l 0 0 0 (2l 1 + 1)(2l 2 + 1)(2l + 1) 4π = l 1 C l 1 l 1 l 1 l 0 0 0 (2l 1 + 1) 2 (2l + 1) 4π δ 0 m m 1 (-1) m 1 l 1 l 1 ℓ m 1 -m 1 0 = l 1 C l 1 l 1 l 1 l 0 0 0 (2l 1 + 1) 2 (2l + 1) 4π 2l 1 + 1δ 0 m δ 0 l ,
in view of the well-known properties ([43, Eq. (8.5.1.1) and Eq. (

)

l 1 l 1 0 0 0 0 = 1 √ 2l 1 + 1 , m 1 (-1) m 1 l 1 l 1 l m 1 -m 1 0 = 2l 1 + 1δ 0 l .
Hence we have, as expected, Ea lm;2 = 0 for all l. Furthermore

E |a lm;2 | 2 = E    l 1 l 2 m 1 m 2 a l 1 m 1 a l 2 m 2 l 1 l 2 l m 1 m 2 -m l 1 l 2 l 0 0 0 (2l 1 + 1)(2l 2 + 1)(2l + 1) 4π l ′ 1 l ′ 2 m ′ 1 m ′ 2 a l ′ 1 m ′ 1 a l ′ 2 m ′ 2 l ′ 1 l ′ 2 l m ′ 1 m ′ 2 -m l ′ 1 l ′ 2 l 0 0 0 (2l ′ 1 + 1)(2l ′ 2 + 1)(2l + 1) 4π    = 2 l 1 l 2 C l 1 C l 2 m 1 m 2 l 1 l 2 l m 1 m 2 -m 2 l 1 l 2 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1)(2l + 1) 4π = 2 l 1 l 2 C l 1 C l 2 l 1 l 2 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1) 4π ,
and the proof is completed.

Remark. Note that V ar T 2 (x) = l 2l + 1 4π C l = 2 l 1 l 2 C l 1 C l 2 (2l 1 + 1)(2l 2 + 1) 4π l 2l + 1 4π l 1 l 2 l 0 0 0 2 = 2 l 1 l 2 C l 1 C l 2 (2l 1 + 1)(2l 2 + 1) (4π) 2 = 2 [V ar {T (x)}] 2 ,
as expected from standard property of Gaussian variables. Here we have used again

l (2l + 1) l 1 l 2 l 0 0 0 2 ≡ 1 .
Our strategy is now the following. We shall first define a very general class, noted D, of quadratic models in terms of the power spectrum of the underlying Gaussian field, and then we shall show that the two notions of HFG and HFE coincide within D.

Definition 8

The centered Gaussian isotropic field T is said to belong to the class D if there exist real numbers α, β such that

1. α ∈ R and β ≥ 0 2. ∞ l=0 l -α+1 e -βl < ∞ 3. there exists constants c 1 , c 2 > 0 such that 0 < c 1 ≤ lim inf l→∞ C l l -α e -βl ≤ lim sup l→∞ C l l -α e -βl ≤ c 2 < ∞ (27)
Remarks.

(1) As a first approximation, the class D contains virtually all models that are relevant for CMB modeling in the case of a quadratic Gaussian subordination. For instance, Sachs-Wolfe models with the so-called Bardeen's potential entail a polynomial decay of the C l (β = 0), whereas the so-called Silk damping effect entails an exponential decay of the power spectrum of primary CMB anisotropies at higher l. We refer again to textbooks such as [START_REF] Dodelson | Modern Cosmology[END_REF], [START_REF] Durren | Cosmology[END_REF] for more discussion on these points.

(2) Note that Condition 2 in the definition of D implies that the parameters α, β must be such that either β = 0 and α > 2, or β > 0 and α ∈ R (with no restrictions).

The next statement is the main achievement of this section. It shows in particular, that the HFG and HFE exhibit the same phase transition within the class D Theorem 9 Let T 2 = H 2 (T ), where the centered Gaussian isotropic field T is an element of the class D. Then, the following three conditions are equivalent

(i) T 2 is HFG (ii) T 2 is HFE (iii) β > 0 and α ∈ R.

Proof of Theorem 9

From [28, Section 6], we already know that Conditions (i) and (iii) in the statement of Theorem 9 are equivalent. The proof of the remaining implication (ii) ⇐⇒ (iii) is divided in several steps.

We start by showing that, if (iii) is not verified, then the angular power spectrum of the transformed field, under broad conditions, exhibits the same behaviour as the angular power spectrum of the subordinating field.

Lemma 10 Suppose β = 0 and α > 2, then

3 × 2 α 4π C l c 2 1 c 2 ≤ C l;2 ≤ c 2 2 c 1 π {2ζ(α -1) + ζ(α)} C l/2 = O(C l ) ,
where ζ(.) denotes the Riemann zeta function.

Proof. We have

l 1 l 2 C l 1 C l 2 l 1 l 2 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1) 4π ≤ 2 l 1 ≤l 2 C l 1 C l 2 l 1 l 2 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1) 4π ≤ 2 c 2 c 1 C l/2 l 1 ≤l 2 C l 1 l 1 l 2 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1)
4π , because (l 1 ∨ l 2 ) > l/2 by the triangle conditions and sup

l 2 ≥l/2 C l 2 /C l/2 ≤ c 2 /c 1 . Now l 1 ≤l 2 C l 1 l 1 l 2 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1) 4π ≤ l 1 C l 1 (2l 1 + 1) 4π l 2 (2l 2 + 1) l 1 l 2 l 0 0 0 2 = l 1 C l 1 (2l 1 + 1) 4π < ∞ .
More precisely

l 1 C l 1 (2l 1 + 1) 4π ≤ c 2 4π l (2l + 1)l -α ≤ c 2 4π {2ζ(α -1) + ζ(α)} . Hence C l;2 ≤ c 2 2 2c 1 π {2ζ(α -1) + ζ(α)} C l/2 .
The upper bound is then established. For the lower bound, it is sufficient to show that

l 1 l 2 C l 1 C l 2 l 1 l 2 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1) 4π ≥ l 2 C 1 C l 2 l 1 l 2 l 0 0 0 2 3(2l 2 + 1) 4π ≥ 32 α C l c 2 1 c 2 l 2 l 1 l 2 l 0 0 0 2 (2l 2 + 1) 4π ≥ 3 × 2 α 4π C l c 2 1 c 2 , as claimed.
Loosely, the previous Lemma 10 states that, under algebraic decay, the rate of convergence to zero of the angular power spectrum is not affected by a quadratic transformation, i.e. C l;2 ≃ C l . The following result holds for fixed l, and it is therefore not related to the high-frequency asymptotic behaviour of the power spectrum {C l } (see [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF] for related computations). Note that we use the notation

C l;2 = 1 2l + 1 l m=-l |a lm;2 | 2 , C l;2 = C l;2 C l;2 .
Lemma 11 Let T 2 be defined by [START_REF] Marinucci | A central limit theorem and higher order results for the angular bispectrum[END_REF]. Then we have

E C l;2 -1 2 = 16 C 2 l;2 l 1 l 2 l 3 C 2 l 1 C l 2 C l 3 l 1 l 2 l 0 0 0 2 l 1 l 3 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1)(2l 3 + 1) (4π) 2 + R(l) ,
where for all l = 1, 2, ....

0 ≤ R(l) ≤ 4 2l + 1 .
Proof. In the sequel, we shall use repeatedly the unitary properties of Clebsch-Gordan coefficients, i.e.

m 1 m 2 l l L m 1 m 2 M l l L ′ m 1 m 2 M ′ = δ L ′ L δ M ′ M 2L + 1 . ( 28
)
Recalling 20, 21, we need to evaluate

1 (2l + 1) 2 C 2 l;2 m 1 m 2 cum {a lm 1 , a lm 1 , a lm 2 , a lm 2 } = 1 (2l + 1) 2 C 2 l;2 m 1 m 2 (-1) m 1 +m 2 cum {a lm 1 , a l,-m 1 , a lm 2 , a l,-m 2 } . Now cum {a lm 1 , a l,-m 1 , a lm 2 , a l,-m 2 } = = cum    l 1 l 2 µ 1 µ 2 a l 1 µ 1 a l 2 µ 2 l 1 l 2 l µ 1 µ 2 m 1 l 1 l 2 l 0 0 0 (2l 1 + 1)(2l 2 + 1)(2l + 1)
4π ,

   = l 1 l 2 l 3 l 4 C l 1 C l 2 C l 3 C l 4 l 1 l 3 l l 4 l 2 l l 1 l 2 l 0 0 0 × × l 3 l 4 l 0 0 0 l 1 l 3 l 0 0 0 l 4 l 2 l 0 0 0 (2l + 1) 2 4 i=1 (2l i + 1) (4π) 2 .
Now by Cauchy-Schwartz inequality and recalling that

l 1 l 3 l l 2 l 4 l ≤ 1 2l + 1
for all l 1 , l 2 , l 3 , l 4 , the previous quantity can be bounded by

1 2l + 1 l 1 l 2 l 3 l 4 C l 1 C l 2 C l 3 C l 4 l 1 l 2 l 0 0 0 l 3 l 4 l 0 0 0 l 1 l 3 l 0 0 0 l 4 l 2 l 0 0 0 (2l + 1) 2 4 i=1 (2l i + 1) (4π) 2 ≤   l 1 l 2 l 3 l 4 C l 1 C l 2 C l 3 C l 4 l 1 l 2 l 0 0 0 2 l 3 l 4 l 0 0 0 2 (2l 3 + 1)(2l 4 + 1) 4π (2l 1 + 1)(2l 2 + 1)(2l + 1) 4π   1/2 ×   l 1 l 2 l 3 l 4 C l 1 C l 2 C l 3 C l 4 l 1 l 3 l 0 0 0 2 l 2 l 4 l 0 0 0 2 (2l 3 + 1)(2l 4 + 1) 4π (2l 1 + 1)(2l 2 + 1)(2l + 1) 4π   1/2 = 2l + 1 4 C 2 l;2 , whence 8 (2l + 1) 2 C 2 l;2 m 1 ,m 2 B(m 1 , -m 1 , m 2 , -m 2 ) ≤ 2 2l + 1 . It is easy to see that m 1 m 2 A(m 1 , -m 1 , m 2 , -m 2 ) = m 1 m 2 C(m 1 , -m 1 , m 2 , -m 2 )
. In view of 20, 21, the statement of the lemma follows easily. The proof of Theorem 9 is now concluded by the following lemma.

Lemma 12 If β = 0 and α > 2, then lim inf l→∞ E C l;2 -1 2 ≥ C 2 2 c 3 2 2c 2 1 π {2ζ(α -1) + ζ(α)} 2 α -2 > 0.
If β > 0 and α is real, then

lim l→∞ E C l;2 -1 2 = 0 .
Proof. For the first part, from Lemma 11 we can focus on

1 C 2 l;2 l 1 l 2 l 3 C 2 l 1 C l 2 C l 3 l 1 l 2 l 0 0 0 2 l 1 l 3 l 0 0 0 2 (2l 1 + 1)(2l 2 + 1)(2l 3 + 1) (4π) 2 = 1 C 2 l;2 l 1 l 2 (2l 1 + 1)(2l 2 + 1)C l 1 C l 2 l 1 l 2 l 0 0 0 2 l 3 C l 1 C l 3 l 1 l 3 l 0 0 0 2 (2l 3 + 1) (4π) 2 , which is larger than 1 C 2 l;2 l 2 (2l 2 + 1)C 2 C l 2 2 l 2 l 0 0 0 2 l 3 C 2 C l 3 2 l 3 l 0 0 0 2 (2l 3 + 1) (4π) 2 ≥ C 2 2 C 2 l+2 C 2 l;2 l 2 (2l 2 + 1) 2 l 2 l 0 0 0 2 l 3 2 l 3 l 0 0 0 2 (2l 3 + 1) (4π) 2 = C 2 2 C 2 l+2 C 2 l;2
. Now we have proved earlier that in the polynomial case, C l;2 ≃ C l ≃ l -α , so the previous ratio does not converge to zero and C l;2 cannot be ergodic; the lower bound provided in the statement of the Lemma follows from previous computations and easy manipulations.

For the second part of the statement, it is sufficient to note that

1 C 2 l;2 l 1 l 2 (2l 1 + 1)(2l 2 + 1)C l 1 C l 2 l 1 l 2 l 0 0 0 2 l 3 C l 1 C l 3 l 1 l 3 l 0 0 0 2 (2l 3 + 1) (4π) 2 ≤ sup l 1 (2l 1 + 1) -1 l 3 Γ l 1 Γ l 3 C l0 l 1 0l 3 0 2 l 1 l 3 Γ l 1 Γ l 3 C l0 l 1 0l 3 0 2 ≤ sup l 1 l 3 Γ l 1 Γ l 3 C l0 l 1 0l 3 0 2 l 1 l 3 Γ l 1 Γ l 3 C l0 l 1 0l 3 0 2
, so the condition is met, just as for the standard case of [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF].

Remarks. (1) By inspection of the previous proof, we note that we have shown how the sufficient condition for asymptotic Gaussianity (HFG) is also such for ergodicity (HFE). More precisely, we have proved that

lim l→∞ sup l 1 sup l 1 l Γ l 1 Γ l 2 C l0 l 1 0l 2 0 2 l 1 l 2 Γ l 1 Γ l 2 C l0 l 1 0l 2 0 2 = lim l→∞ sup λ P (Z 1 = l 1 |Z 2 = l 2 ) = 0 ,
where {Z l } is the Markov chain defined in [28, Eq. ( 57) and (58)] is a sufficient condition for the HFG (see [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF]Proposition 9]) and also a sufficient condition to have lim l→∞ E C l -1 2 = 0 . (2) In principle, the case q = 3 can be dealt along similar lines.

(3) (On Cosmic Variance) Loosely speaking, the epistemological status of Cosmological research has always been the object of some debate, as in some sense we are dealing with a science based on a single observation (our observed Universe). In the CMB community, this issue has been somewhat rephrased in terms of so-called Cosmic Variance -i.e., it taken as common knowledge that parameters relating only to lower multipoles (such as the value of C l , for small values of l) are inevitably affected by an intrinsic uncertainty which cannot be eliminated (the variability due to the peculiar realization of the random field that we are able to observe), whereas this effect is taken to disappear at higher l (implicitly assuming that something like the HF E should always hold). Our result seem to point out, apparently for the first time, the very profound role that the assumption of Gaussianity may play in this environment. In particular, for general non-Gaussian fields there is no guarantee that angular power spectra and related parameters can be consistently estimated, even at high multipoles -i.e., the Cosmic Variance does not decrease at high frequencies for general non-Gaussian models.

Discussion and directions for further research

This paper leaves many directions open for further research. We believe the results of the previous two sections point out a very strong connection between conditions for High Frequency Ergodicity (HFE) and High Frequency Gaussianity (HFG) for isotropic spherical random fields. It is natural to suggest that equivalence may hold for Gaussian subordinated fields of any order q, or even more broadly for general Gaussian subordinated fields on homogeneous spaces of compact groups. Indeed, in this broader framework it is shown in [START_REF] Baldi | On the characterization of isotropic random fields on homogeneous spaces of compact groups[END_REF] that independence of Fourier coefficients implies Gaussianity, which is the heuristic rationale behind our results here.

The connection between the HFE property can also be studied under a different environment than Gaussian subordination. Consider for instance the class of completely random spherical fields, which was recently introduced in [START_REF] Dennis | Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere[END_REF][START_REF] Dennis | Correlations between Maxwell's multipoles for Gaussian random functions on the sphere[END_REF]. Following the definition therein, we shall say that a spherical random field is completely random if for each l we have that the vector a l. = (a l,-l , ..., a ll ) is invariant with respect to the action of all matrices belonging to SU (2l + 1) and verifies a lm = (-1) m a lm . Because of this, the vector a l. is clearly uniformly distributed on the manifold of random diameter |a lm | 2 = C l , or equivalently, introducing the (2l + 1) vector U l

U l = 1 √ 2l + 1    √ 2Re a l1 C l , √ 2Re a l2 C l , ...., a l0 C l , √ 2Im a l1 C l , ..., √ 2Im a ll C l    (29)
it holds that, for l large, it holds approximately that U l law ∼ U (S 2l ), i.e. U l it is asymptotically distributed on the unit sphere of R 2l+1 . Under these conditions, it is simple to show that HF E ⇒ HF G, i.e. because the left hand side can be viewed as the marginal distribution for a uniform law on a sphere of growing dimension; the latter is asymptotically Gaussian, as a consequence of Poincaré Lemma (see [START_REF] Diaconis | A dozen de Finetti-style results in search of a theory[END_REF]). We do not investigate this issue more fully here, and we leave for future research the determination of general conditions such that (compare with (29))

the law of U l and U (S 2l ) are asymptotically close as l → ∞ .

Obviously, for all fields such that (30) holds (i.e. those that are asymptotically completely random, to mimic the terminology of [START_REF] Dennis | Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere[END_REF][START_REF] Dennis | Correlations between Maxwell's multipoles for Gaussian random functions on the sphere[END_REF]), by the same argument as before we have that

   C l C l → prob 1    ⇒ T l (2l + 1)C l law
→ N (0, 1) .

To conclude this work, we wish to provide an example were the HFE and HFG property are indeed not equivalent. Consider the (anisotropic) field Note that, in the definition of h (x), the sum is not taken with respect to l. The field can be made isotropic by taking a random rotation T (x) = h(gx), where g is a random, uniformly distributed element of SO(3). We have as usual T (x) = l l m=-l a lm Y lm (x) , where

a lm law = l m ′ =-l D l m ′ m (g)ξ lm ′ law = 4π 2l + 1 Y lm (g)ξ l ,
and where D l (g) denotes the well-known Wigner representation matrices for SO(3), and the first identity in law is discussed for instance in [START_REF] Baldi | Some characterizations of the spherical harmonics coefficients for isotropic random fields[END_REF], [START_REF] Marinucci | Representations of SO(3) and angular polyspectra[END_REF]. Note that ≡ 0, while asymptotic Gaussianity fails. Hence, we have constructed an example where the HFE property holds but the HFG property does not. Note that the support of the vector {a l. } is concentrated on a small subset of the sphere S 2l ; heuristically, this is what prevents Poincarè-like arguments to go through.

w

  1l (L) = 2l L=0 w 2l (L) = 1 .

l 3 l 4 µ 3 µ 4 a l 3 µ 3 a l 4 µ 4 l 3 l 4 l µ 3 µ 4 -m 1 l 3 l 5 l 6 µ 5 µ 6 a lµ 5 a lµ 6 l 5 l 6 l µ 5 µ 6 m 2 l 5 l 7 l 8 µ 7 µ 8 a

 4136258 lµ 7 a lµ 8 l 7 l 8 l µ 7 µ 8 -m 2 l 7 l 8 l 0 0 0 (2l 7 + 1)(2l 8 + 1)(2l + 1) 4π

→

  N (0, 1) , as l → ∞ . Indeed, it is sufficient to note that, as before T l (x)

h

  (x) = lm ξ lm Y lm (x) , where ξ lm = ξ l , for m = 0 0 , otherwise , and the random variables ξ l verifies the assumption Eξ l = 0, l Eξ 2 l < ∞ and Eξ 4 l < ∞ .

2 = ξ 2 l 2 l 1 and indeed for general sequences {ξ l } E ξ 2 lHowever, in the special case where ξ l = e -l with probability 1 2 -e -l with probability 1 2 ,

 2221222 |Y lm (g)| 2 = ξ 2 l , as expected, because the sample angular power spectrum is invariant to rotations. Of course in this case we do not have ergodicity in general, i.e. it may happen that l m=-l |a lm | 2 E l m=-l |a lm | Eξ Eξ

and counting equivalent permutations = 8

For the first term, note first that (-1) m 1 +m 2 +µ 1 +µ 2 +µ 3 +µ 4 ≡ 1, because the exponent is necessarily even by the properties of Wigner's coefficients. Moreover, applying iteratively [START_REF] Marinucci | Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, Bernoulli[END_REF]

Likewise, for the second term we note that (-1) µ 1 +µ 2 +µ 3 +µ 4 ≡ 1, and using [START_REF] Marinucci | High-resolution asymptotics for the angular bispectrum of spherical random fields[END_REF] m 1 m 2 (-1) m 1 +m 2 B(m 1 , -m 1 , m 2 , -m 2 )


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Abdelghani May

Effect Of Cooling Rate During Heat Treatments On The Cyclic Behavior And Fatigue Damage Of 2017A Aluminum Type Alloy

Keywords: ALUMINUM ALLOY, HEAT TREATMENT, CYCLIC BEHAVIOR, FATIGUE DAMAGE, COOLING RATE EFFECT

This study investigates the effects of cooling rate during heat treatments on the cyclic behavior and fatigue damage of 2017A aluminum type alloy. First, monotonic tensile tests were performed to quantify the evolution of the mechanical properties of the 2017A alloy. The obtained results allow us to observe that the nitrogen liquid quenching slightly increased the mechanical characterizations compared to the ice water quenching due to the quench sensitivity. With regard to the fatigue behavior study, cyclic strain imposed tension-compression loading tests at an amplitude of 0.3% were carried out. The obtained results show that the 2017A alloy demonstrates a cyclic hardening phenomenon for homogenized state up to failure, whereas the other states show a hardening up to cycle 100 and then a stabilization until rupture. In addition, the fatigue life is strongly affected by the cooling rate during quenching. Indeed, fatigue life is inversely proportional to the quenching temperature. However, the results have shown that by increasing the temperature the fatigue life increases also, but when reaching a maximum tempering temperature the fatigue life decreases. Scanning Electron Microscope (SEM) observations demonstrated that the dominant type of rupture is ductile. Moreover, considering that the failure is always in the maximum slip direction, we have deduced that the failure mode of the specimens is mode I.

INTRODUCTION

Aluminum alloys are widely used in the automotive and aerospace industries due to their high mechanical strength combined with low density and good fracture toughness. The high-strength aluminum alloy considered in the present study belongs to the Al-Cu alloys family. Such alloys were specifically designed for their useful mechanical properties in critical structural parts of aircraft. It contains copper (Cu) as a major alloying element and its hardening is obtained by thermal aging [START_REF] Khan | Experimental and numerical lifetime assessment of Al 2024 sheet[END_REF]. Indeed, heat treatments play a very important role in improving the mechanical of these alloys. A homogenization treatment, tempering and over-tempering, is commonly practiced for these alloys. The hardenability results from the interaction between the alloying elements and the applied heat treatment. It is well-known that the hardening of this alloy arises from Al2Cu and Al2CuMg precipitation, provided that the particles are finely and densely distributed [START_REF] Lados | Relationships between microstructure and fatigue crack propagation paths in Al-Si-Mg cast alloys[END_REF][START_REF] May | Analysis of the cyclic behavior and fatigue damage of extruded AA2017 aluminum alloy[END_REF][START_REF] Wang | Fatigue and fracture behaviour of structural Al-alloys up to very long life regimes[END_REF].

This work aimed to establish the heat treatment parameters and to progress in the understanding of the influence of this treatment on the cyclic fatigue behavior of alloy 2017A. Homogenization, rapid quenching and tempering heat treatments were carried out with control of the cooling rate by two types of quenching, one with ice-cold water and the other with liquid nitrogen. The influence of these treatments on the mechanical properties and fatigue behavior was evaluated at the macroscopic and microscopic scale.

EXPERIMENTAL PROCEDURE

Material and specimen preparation

The studied material is an AA2017 aluminum alloy with the average chemical composition given in Tab. To ensure the same microstructural state of the material and therefore the same initial mechanical properties, all specimens were heat treated according to homogenization annealing thermal cycle at 500°C for 6 h. The followed heat treatment consist to realize a structural hardening of specimens in three steps (Fig. 1): a solutionizing treatment at 500°C for 20 min, rapid quenching in salted ice water (0°C) or in liquid nitrogen (-200°C), and finally maturation or aging during 24 h at different temperatures (120°C, 180°C and 240°C). Finally, they were stored at -14°C to maintain the metallurgical state of the samples until the beginning of the test.

Description of mechanical tests

To investigate the heat treatment effect on the fatigue behavior of 2017A aluminum alloy, two series of mechanical tests were carried out. In the first series, monotonic tensile tests were performed in order to identify the effect of thermal quenching in ice water or in liquid nitrogen on the mechanical characteristics of this alloy.

In the second series, fully reversed tension-compression fatigue tests were carried out in strain controlled cycles with a strain amplitude equal to ± 0.3% and a frequency of 03 Hz on the specimens subjected to the various stages and aging temperature of the heat treatment described above. The obtained lifetimes and stress response are compared to evaluate the heat treatments effect on the fatigue behavior.

EFFECT OF HEAT TREATMENT ON MONOTONIC BEHAVIOR UNDER TENSILE TESTS

The tensile tests results (conventional tensile curves) are presented in Fig. 2, from these curves; we can show the mechanical properties of the 2017A alloy under different states of heat treatments. Given the large number of the studied heat treatment states and in the objective of readability and comparison, it was considered necessary to present only the homogenization and quenching treatment states curves. Tab. 2 presents the mechanical properties of the 2017A alloy under different treatment states.

The observation of the tensile curves allow highlighting the cooling rate effect on the yield strength (elastic limit), the tensile strength and the elongation. Indeed, the quenched specimens showed an increase in mechanical properties compared to those with homogenization stage only. In addition, we noticed that the nitrogen liquid quenching slightly increased the mechanical characterizations compared to the ice water quenching due to the quench sensitivity. However, we have observed that Young's modulus is approximately identical for the different treatment states.

EFFECT OF HEAT TREATMENT ON THE CYCLIC BEHAVIOR AND FATIGUE LIFE

As explained in the previous section, the effect of heat treatment on fatigue life has been investigated through the realization of fatigue tests under strain-controlled tension-compression in an amplitude of ± 0.3% on specimens initially subjected to the heat treatment described above. To give a first overview of the elastoplastic behavior evolution of the 2017A alloy in fatigue tests, the stress-strain loops comparison for the different metallurgical states are represented in the Fig. 3. The analyzed results in terms of the evolution of the stress amplitude show that the heat treatments have an effect on the behavior of the material studied (2017A) in cyclic loading. The cyclic behavior tests show a phenomenon of cyclic hardening for the homogenized state until failure, on the other hand the other states show a hardening until cycle 100 then a stabilization until failure (Fig. 4). However, the observed stress levels did not show a significant difference for the studied metallurgical stats. The values of the stresses remain more or less close, in particular during the quasi-plateaus at mid-life.

In order to evaluate the heat treatments effect on cyclic lifetimes in fatigue of the 2017A alloy, Table 3 provides a summary and comparison of the lifetimes obtained for each metallurgical stats. We find that quenching with ice water increases the fatigue lifetimes by a factor of 2 (~3100 cycles) compared to the homogenized state.

In opposition, the liquid nitrogen quenching was damaging (~1000 cycles) compared to the homogenization metallurgical state (1500 cycles), which can, due to excessive cooling, causes distortion or deformation in the specimens, and causes a very high level of internal stress. Furthermore, it can also be observed that the more the tempering temperature increases, the more the fatigue lifetimes also increases until reaching a maximum temperature (180°C) where it decreases. In order to understand the mechanisms of 2017A alloy failure under the applied treatments effect, microstructural analyzes were carried out using a Scanning Electron Microscope (SEM) on the fracture faces of the tested specimens (Fig. 5). It has been observed that the dominant type of rupture in this material is ductile. Indeed, the investigation of the morphologies of the rupture zones for all the specimens shows the presence of cupules. Inside the latter, precipitates remain stuck and propagation generally occurs in three sequences : (i) germination of cavities around inclusions, (ii) cavity growth and (iii) coalescence of the cavities leading to the appearance of a macroscopic crack. These cupules lead to the reduction of the active surface of the specimen until the rupture, consequently a ductile rupture occurs. Moreover, considering that the failure is always in the maximum slip direction, we have deduced that the failure mode of the specimens is mode I.

CONCLUSIONS

The main objective of this work was to establishment of the heat treatments parameters, and to progress in understanding the influence of this treatment on the monotonic and the cyclic fatigue behavior of the 2017A alloy. We have found that using liquid nitrogen as a quenching liquid allows obtaining better mechanical characteristics (σy, σmax) compared to ice water in the case of monotonic characterization. However, the cyclic behavior tests showed that the 2017A alloy demonstrates a cyclic hardening phenomenon for homogenized state up to failure, whereas the other states show a hardening up to cycle 100 and then a stabilization until rupture. Moreover, the ductility drops drastically with the increase of the cooling rate, and it can be confirmed that the tempering temperature is proportional to the mechanical characteristics, until it reaches a critical temperature (threshold), where we have found that this proportionality is reversed. In the same way, obtained fatigue lifetimes confirmed that quenching with ice water showed an increase in lifetimes compared to the homogenized state, contrary to the liquid nitrogen quenching, due to excessive cooling. Scanning Electron Microscope (SEM) observations on the fracture faces has been demonstrated that ductile failure was dominant type for this material. Moreover, considering that the failure is always in the maximum slip direction, we have deduced that the failure mode of the specimens is mode I.

Fig. 1 .

 1 Fig. 1. Summary of the applied heat treatments.

Fig. 2 .Tab. 2 .

 22 Fig. 2. Conventional tensile curves of the 2017A alloy under the different treatments states.

Fig. 3 .

 3 Fig. 3. Comparison at mid-life of hysteresis loops of fatigue tests: (a) homogenization vs. quenching, (b) quenching in ice water and (c) quenching in nitrogen.

Fig. 4 .

 4 Fig. 4. Comparison of stress amplitude evolution of fatigue tests: (a) homogenization vs. quenching, (b) quenching in ice water and (c) quenching in nitrogen.

Fig. 5 .

 5 Fig. 5. Fracture faces SEM observations : (a) homogenization, (b) quenching in ice water (1) and (c) quenching in nitrogen (2), (d) (1) + tempering 180 °C and (e) (2) + tempering 180 °C.

1. Tab. 1.

  Average chemical composition of AA2017 aluminum alloy in % used in the study.

	Cu	Mg	Mn	Fe	Si	Zn	Ni	Cr	Al
	3.9-4.0	1.1-1.2	0.7-0.8	0.5-0.6	0.6-0.7	<0.21	<0.16	<0.10	rest

  Tab. 3. Fatigue lifetimes obtained for each metallurgical stats (Cycles).

	Metallurgical stats	Test 1	Test 2	Test 3	Average
	Homogenization annealing	1700	1500	1400	1533
	Quenching in ice water (1)	3200	3600	2600	3133
	(1) + Tempering 120°C	3300	3500		3600
	(1) + Tempering 180°C	5300	5600	5500	5467
	(1) + Tempering 240°C	4000	4800	4900	4567
	Quenching in liquid nitrogen (2)	500	1600	800	967
	(2) + Tempering 120°C	2600	2300	2100	2333
	(2) + Tempering 180°C	3100	inconclusive tests due to a	3100
	(2) + Tempering 240°C	2800	large dispersion	2800