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Asymptotic Behaviour of Level Sets of Needlet Random Fields

We consider sequences of needlet random fields defined as weighted averaged forms of spherical Gaussian eigenfunctions. Our main result is a Central Limit Theorem in the high energy setting, for the boundary lengths of their excursion sets. This result is based on Stein-Malliavin techniques and Wiener chaos expansion for nonlinear functionals of random fields. To this end, a careful analysis of the variances of each chaotic component of the boundary length is carried out, showing that they are asymptotically constant, after normalisation, for all terms of the expansion and no leading component arises.


Introduction and Main Result
Spherical random fields constitute one of the central subjects in random geometry. It has been extensively studied in recent years not only due to its theoretical interest but also because of its importance in many applied domains, particularly in physics. Let us consider {f (x), x ∈ S 2 }, an isotropic Gaussian spherical random field with zero mean. It is known that f (x) can be represented, in the L 2 -sense, as where {a m } are Gaussian random variables satisfying E[a mā m ] = C δ δ m m and {Y m } are the spherical harmonics (see for instance [16], [15], [14]). The sequence {C } is called angular power spectrum of f (x) and is such that C 2 +1 4π = Ef 2 < ∞ (see [17]). The random fields {f (x), x ∈ S 2 } are the eigenfunctions of the Laplace-Beltrami operator and hence they satisfy the Helmholtz equation ∆ S 2 f + ( + 1)f = 0, ∈ N. They, too, are isotropic centered Gaussian with covariance function given by where P is the Legendre polynomial of degree and d(x, y) is the spherical distance between x and y on S 2 . The eigenfunctions {f } have been widely investigated by many authors, in particular their excursion sets have been considered, for example, in [32], [33], [20], [21], [19], [6]. However, in many practical situations, such as statistics for spherical observations, a Fourier analysis approach is not optimal if the data exhibits some blind spots. In this case procedures based on wavelet constructions on the sphere are preferred, in view of their double localization properties in real and harmonic space. Wavelet systems find applications, for instance, in astrophysics and cosmology (see for example [23], [7], [9], [18], [27], [22]). They are used to extract information from spherically observed signals in these fields, since the presence of a masked region in the domain of observation does lower the efficiency of the analysis. Spherical wavelet systems, so called needlets, have been introduced by [24], [25] and investigated in the last years by many authors, see for instance [4], [12], [31], [11], [3], [5], [10]. In particular in [3], [5], [10] the properties of needlets applied to random fields are studied. This paper aims to give a contribution in this direction.
To this purpose we consider an averaged form of the functions f (x), defined as for j ∈ N. Here B > 1 is a fixed parameter, called bandwidth, and b(· ) is a C ∞ function with compact support on [1/B, B] and satisfies the property j∈N b B j 2 = 1 for all > 2 (see [16]).
Let us consider the needlet kernel, as introduced by [24], defined for any j = 1, 2, ..., as follows Then the random fields β j (x) can be seen as spherical needlets coefficients: The localization property of Ψ j (x, y), established in [24], is given by a bound which is nearlyexponential on B j , j ∈ N. More precisely, we have that for all (x, y) ∈ S 2 and for all integers M, there exists a constant C M such that This property enables us to find a useful bound for the covariance function of the random field β j (x) (see Theorem 13.1 [16] and [3]).
In [5] the authors investigated the asymptotic behaviour of needlets polyspectra, defined as where H q is the Hermite polynomial of order q. It turned out that the above integral has the same rate of convergence for each order q. Moreover, since the area of the excursion sets can be expanded in the L 2 -sense through the Hermite polynomials (see also [21] for details), applying some Stein-Malliavin results (the theory being detailed, for example, in [26]), the authors obtained a quantitive Central Limit Theorem (CLT) for the excursion area.
The purpose of our paper is to advance in the study of Lipschitz-Killing curvatures for needlet random fields. We investigate the boundary length of excursion sets of the normalised random fieldβ j (x) := β j (x)/ E[β j (x) 2 ], defined as L j (z) = {x ∈ S 2 :β j (x) = z}, (1.3) where z ∈ R is a fixed level. As for the area, the length L j (z) can be expanded, in the L 2 − sense, in terms of its q-th order chaotic components to obtain the orthogonal expansion proj[L |C q ], denoting the projection on a subspace of L 2 called qth Wiener chaos (see Section 3 and [28], [20] for more details). The projection proj[L j (z)|C q ] involves integrals of Hermite polynomials of the form for u = 0, . . . , q, k = 0, . . . , u; where x = (x 1 , x 2 ) ∈ S 2 , (1.5) for i = 1, 2 and ∂ i f (x) := ∂ ∂xi f (x) (see [20] for details) and∂ iβj (x) being its renormalised version. The study of these integrals extends the findings on the polyspectra from [5] that emerge as a special case for k = u = 0. After a careful analysis of the variance of each chaotic component of L j (z) we establish a Central Limit Theorem (CLT). Theorem 1.1. Let L j (z) be defined as in (1.3), we have that, as j → ∞, where N ∼ N (0, 1) and d W denotes the Wasserstein distance.
Note that since proj[L j (z)|C q ] is a sequence inside a fixed Wiener chaos (see Section 3), the celebrated Fourth Moment Theorem allows to establish a CLT for any single component. When dealing with the boundary length of the excursion sets for the eigenfunctions f one can show that there is only a single leading component in the chaos expansion. In view of this the proof of the CLT for the whole series is immediate and was carried out in [28]. This is not the case in our framework. Indeed, up to possible cancellation of the constant for discrete specific values of the level z, the variance of proj[L j (z)|C q ] is asymptotically constant for each q after normalisation. This analysis has been carried out differently for points far from the diagonal and for the ones close to it. For the latter we use an argument based on the two-point correlation function (see [32]) showing that here the variance is finite. For points far from the diagonal we are able to prove that all the terms of the expansion have the same rate of convergence, using the Wiener chaos expansion and exploiting Hilb's asymptotic results. Therefore, no shortcut is possible in the study of the CLT and the entire series must be considered, similarly to the excursion area studied in [5]. Finally, making use of some Stein-Malliavin results (Theorem 5.1.3 [26]) and needlets localization properties, we prove Theorem 1.1.

Plan of the paper
In Section 2 we introduce the main objects of the work, namely needlet random fields and their derivatives and derive useful results about their covariance functions. In Section 3 we provide some background on Wiener chaoses and on the Stein-Malliavin method which is essential for the understanding of the paper. Section 4 investigates the asymptotic variance of the boundary lengths for needlet coefficients, showing the asymptotic behaviour of every single Wiener chaos component of the chaotic expansion of the considered geometric functional. Finally the CLT is proved in Section 5. In the Appendix all the technical tools exploited in the main proofs are collected. More precisely, in 6.1 we report and derive some properties of needlet kernels; in 6.2 other auxiliary results are recalled; 6.3 and 6.4 contain the proofs of some technical lemmas and propositions used in the study of the variance and the Central Limit Theorem, respectively.

Needlet random fields and their derivatives
We start by considering the objects defined in (1.1) and (1.5); the bandwidth parameter from now on will be B = 2. We first normalize all the random fields to have unit variance, hence, let us denoteβ Moreover, since the random fields f (x) are uncorrelated for different we get and substituting E[ (see for example [28]) we obtain Similarly to [28], it can be seen that also E[(∂ 2βj (x)) 2 ] is equal to A 2 j . Then we definẽ .
Let us now compute the covariance functions involved in this setting which we will exploit in the next sections. By isotropy permitting to fix the point x = N , where N denotes the North Pole, and using spherical coordinates (θ, ϕ) ∈ [0, π] × [0, 2π), it was shown that for all x, y ∈ S 2 , where θ denotes the angle between x and y (see [20] for details), which leads tõ In order to investigate the asymptotic behaviour of the boundary length for needlet random fields we assume some standard regularity conditions on the power spectrum C (see [16], page 257).
Condition 2.1. There exists M ∈ N, a > 4 and a sequence of functions {g j (·)} such that for where c −1 0 ≤ g j ≤ c 0 for all j ∈ N and for some c 1 , . . . , c M > 0 and r = 1, . . . , m we have Condition 2.1 is fulfilled for example by models of the form C = −a G( ) where G( ) = P ( )/Q( ) and P ( ), Q( ) > 0 are two positive polynomials of the same order.
For such power spectrum models it was shown in [5] that where G is the limit of G( ) for → ∞. In the following lemma we show that under the assumption of Condition 2.1 a similar asymptotic result holds also for A 2 j .
Proof. Note that The result now follows by dominated convergence.
In the same way as in Lemma 2.1 it can be proved that for p ∈ R:

Malliavin calculus
In order to prove a Central Limit Theorem we will use results from Malliavin calculus established for functionals of Gaussian random processes. Below we give a brief overview over the main definitions and methods that will be applied (closely following [26]) and transfer the theory to our setting.
Let H be a real separable Hilbert space with its associated inner product ., . H , and (W (ϕ), ϕ ∈ H) an isonormal Gaussian process on a probability space (Ω, F, P ), which is a centered Gaussian family of random variables such that E (W (ϕ)W (ψ)) = ϕ, ψ H for every ϕ, ψ ∈ H.
Note that for H = L 2 (A, A, µ) over a Polish space A with the associated σ-field A and a positive σ-finite and non-atomic measure µ one can define an isonormal Gaussian process with respect to the inner product g, h H = A g(a)h(a)dµ(a) as the Wiener-Itô integral Denote by I q the qth multiple stochastic integral with respect to W , that is, an isometry between the Hilbert space H q (meaning symmetric tensor product) equipped with the norm 1 √ q! · H ⊗q and the Wiener chaos C q of order q, which is defined as the closed linear span of the random variables H q (W (ϕ)), ϕ ∈ H with ϕ H = 1 and H q the Hermite polynomial of degree q ≥ 1 defined by: For f ∈ H such that f H = 1 we have for every q ≥ 1 H q (W (f )) = I q (f ⊗q ) and the isometry property can be written as follows: for p, q ≥ 1, f ∈ H ⊗p and g ∈ H ⊗q wheref denotes the canonical symmetrization of f . The space L 2 (Ω, F, P ) can be decomposed in terms of Wiener chaoses: Every F ∈ L 2 (Ω, F, P ) admits a unique expansion in L 2 -sense Another definition that we will need is that of contraction on H = L 2 (A, A, µ) For f ∈ H ⊗p , g ∈ H ⊗q the rth contraction f ⊗ r g is the element of H ⊗(p+q−2r) which is defined by (f ⊗ r g)(s 1 , . . . , s p−r , t 1 , . . . , t q−r ) = A r f (s 1 , . . . , s p−r , u 1 , . . . , u r )g(t 1 , . . . , t q−r , u 1 , . . . , u r )dµ(u 1 ) . . . dµ(u r ) for every r = 1, . . . , p ∧ q.
We have the following product formula for multiple integrals: if f ∈ H p and g ∈ H q , then Let us denote by D the Malliavin derivative operator that acts on cylindrical random variables of the form F = g(W (ϕ 1 ), . . . , W (ϕ n )), where n ≥ 1, g : R n → R is a smooth function with at most polynomially growing derivatives and ϕ i ∈ H. This derivative is an element of L 2 (Ω, H) and it is defined as This operator can be considered as inverse to the multiple integrals in the sense that for q ≥ 1 and The closure of the space of cylindrical random variables with respect to the norm is denoted by D 1,2 .
A further operator that we need to consider is the Ornstein-Uhlenbeck operator L.
The main tool for proving the Central Limit Theorem in this article is the following statement that generalises the celebrated Fourth Moment Theorem.  [26]). Let F ∈ D 1,2 be centred and such that E[F 2 ] = σ 2 > 0. Then for N ∼ N (0, σ 2 ) we have where d W (F, N ) is the Wasserstein distance between F and N .
Consider the Hilbert space L 2 (S 2 , dσ(x)). As explained above, this space can be associated with an isonormal Gaussian process W such thatβ j (x) can be expressed as an integral of the kernelΘ The random variables∂ 1βj (x) and∂ 2βj (x) are almost sure limits of elements of {W (h), h ∈ L 2 (S 2 , dσ(x))} and showing that they are also L 2 limits goes back to demonstrating that the integral and the differentiation in the deterministic formulas are interchangeable. This can be easily seen by verifying this claim for Legendre polynomials. Consequently, we can write∂ 1βj (x) and∂ 2βj (x) as integrals with respect to W of kernels Θ (1) Following a standard argument (explained for example in [2]), one can express the boundary length of level sets (for some level z) ofβ j as the almost sure limit of the integral which equals (with the normalisation from above) Following the same line of argument as was presented for f in [28], we obtain that the convergence also takes place in L 2 .
Relying on the fact that all random variables above are normed and subsequently using the same expansion and approximation as in [28], we arrive at the Hermite expansion of the level sets in terms ofβ j and its partial derivatives: for an even pair of indices and zero otherwise and with the standard Gaussian probability distribution function φ and the -th Hermite polynomial H .

Variance asymptotics of L j (z)
In this chapter we analyse the variance of L j (z) in order to establish the proper normalisation for the Central Limit Theorem. The main object of our study will be in particular We use different methods depending on the relative position of the points x and y in this expression: we handle the case where they are close to each other using the so called 2-point correlation function, and for the case where they are further apart we use the Wiener chaos decomposition and consider each summand separately. In fact, obtaining the order of E[L j (z) 2 ] with respect to j is rather straightforward: we calculate precisely the variance of proj(L j (z)|C 2 ) and show with a simple argument that the variances of all other chaos components are bounded by it. As mentioned in the introduction, in the manuscript [28] in order to prove the Central Limit Theorem for the level sets of the random fields f , the author shows that the second chaos is dominating all other chaos components for z = 0 in the L 2 -sense, making it the only factor contributing to the CLT. As this simplifies the proof, it makes sense to consider applying the same tactics in our case. However, this chapter also includes a result stating that (up to possible cancellation phenomena for specific values of z) every summand of every chaos component exhibits the same rate of convergence with respect to j and must therefore be included in the final analysis. Moreover, this result helps analyse the variance more precisely.
We start with a variance calculation for the first and the second chaos component.
which is nonzero for all z.
Proof. We have by properties of spherical harmonics.
For q = 2 there are three summands for which the constant α k, u−k is nonzero: the one with u = k = 0, with u = 2, k = 0, and with u = k = 2. This translates to and we can write Since A 2 j is of order (2 j ) 2 , for obtaining the convergence result it suffices to show that all four summands are of order 2 −2j . Let us begin with S 1 . We write by the diagram formula (see Proposition 6.2 in the appendix) and (2.2) by orthogonality. As one can see by Remark 2.3, this term is of order 2 −2j . Similarly, for S 2 we obtain with P m n denoting the associated Legendre functions. Note that for two such functions of the same degree m an orthogonality relation holds, namely Therefore, we have This term is also of order 2 −2j . For S 3 we calculate by Proposition 6.3. This term is of order 2 −3j . For S 4 note first that by the characterising differential equation for Legendre polynomials we have Moreover, due to the identity xP (x) = P +1 (x) − ( + 1)P (x) we obtain the identity Therefore, the integral can be decomposed as The first summand equals ( 1 + 1) 2 ( 2 + 1) 2 2 2 1+1 δ 1 2 by orthogonality, the second and third can be bounded (in absolute value) by ( 1 + 1) 2 and ( 2 + 1) 2 respectively by Proposition 6.3, and the last summand's absolute value is bounded by 2( 1 + 2) due to the same proposition. The remaining calculations for S 4 are now straightforward: by the usual change of variables. The first of the four summands arising from the decomposition of this integral indicated above equals and it is of order 2 −2j . The other three summands have bounds that converge even faster. We can calculate the exact limiting variance of the second chaos component by calculating the limits of S 1 , S 2 and the slowest summand of S 4 . For a level z ∈ R it equals which is clearly nonzero for all z.
To analyse the variances of higher chaoses we will need a more general application of the diagram formula for dealing with the expression In this context we have the following lemma. Moreover, Proof. By diagram formula the term Note that, as shown in [20], other summands appearing in the diagram formula do not appear here, since Let us determine the constant M αβγ and understand how this expression can be simplified.
By diagram formula the number M αβγ corresponds to the number of diagrams with 6 rows with q − u 1 , k 1 , u 1 − k 1 , q − u 2 , k 2 and u 2 − k 2 vertices respectively, satisfying the following conditions: Since the second and fifth rows appear only in the condition (v), the number of vertices in these rows must be equal (u 1 − k 1 = u 2 − k 2 ) and they must be connected to each other. The conditions (ii) and (iii) imply β 1 = q − u 1 − α, β 2 = q − u 2 − α, and with condition (iv) we conclude that In other words, all the powers in the formula are explicitly determined by α, so the triple sum becomes a sum over α.
Note that for such a diagram to exist α must lie between max(q − k 1 − u 2 , 0) and min(q − u 1 , q − u 2 ). Let us assume that min(q − u 1 , q − u 2 ) = q − u 2 without loss of generality. Let us now fix an α in this interval and calculate the number of possible diagrams for this α. There are q−u1 α and q−u2 α possibilities to pick α vertices from the first and fourth rows respectively, there are α! possibilities to connect these vertices. There are k2 q−u1−α and k1 q−u2−α choices of β 1 and β 2 vertices respectively as well as (q − u 1 − α)! and (q − u 2 − α)! permutations for each. Additionally there are (k 1 − (q − u 2 − α))! connections of γ and (u 1 − k 1 )! connections of vertices between the second and the fifth rows. In total, we have Let us compute α M α . Recall that this sum ranges from q −k 1 −u 2 to q −u 2 (with the convention n −k = 0 for k ∈ N). We obtain by shifting the index and using the identity By the Chu-Vandermonde identity the above sum over α equals q−u2+k2 k1 , and therefore, using, again, the identity u 1 − k 1 = u 2 − k 2 .
The following proposition establishes a bound on asymptotic variance for all chaos components and answers the question of normalisation for the CLT.
with respect to j.

Proof. We have
Since k 1 , k 2 , u 1 , u 2 are even (otherwise the factor in front of the integral equals zero), the powers β = 2q − u 1 − u 2 − 2α and δ = u 1 − k 1 are even as well. This means that at least one of the integers α, β, γ and δ, whose sum is q, is greater or equal than 2. If α ≥ 2 we bound |E[β j (x)∂ 1βj (y)]| β , |E[∂ 1βj (x)∂ 1βj (y)]| γ and |E[∂ 2βj (x)∂ 2βj (y)]| δ by 1 (recalling that all these covariances are bounded by 1 due to normalisation) and use This is bounded (up to a constant) by S 1 from Theorem 4.1. Similarly, if β, γ or δ is greater than 1 the integral is bounded by S 2 , S 3 or S 4 respectively. By Theorem 4.1 all of these yield summands of order O(1).
Let us now turn to the asymptotics for the case that the integrands are close to each other. This proof can be found in the appendix.
where = 2 j behaves as O(1) as j tends to infinity.
The next two results establish the exact asymptotics of integrals involved in higher chaoses. The proof of the first proposition is technical and will also be carried out in the appendix. Proposition 4.6. For q = 3 and q ≥ 4 we have: Assuming that the constant c q defined in [5] is nonzero, the limiting variance lim j→∞ E[proj(L j (z)|C q ) 2 ] is zero for at most q values of z, i.e. these terms do not converge faster than the variance of the second chaos.
Proof. The limit of is by definition of β · equal to φ(z) 2 multiplied by a polynomial in z of order 2q. Since it is nonnegative, the polynomial does not have any simple zeroes. The only case in which it has more than q zeroes is when all its coefficients are equal to zero. However, we know that its leading coefficient is the factor in front of β q (z) 2 , contained in the summand with u 1 = u 2 = k 1 = k 2 = 0. The factor in front of the integral in this summand is nonzero, and the integral itself is, up to a positive constant, exactly c q from [5]. Under the assumption of this proposition it is nonzero.

Central Limit Theorem
In this section we finally prove the Central Limit Theorem for the boundary length of excursion sets of needlet random fields. The proof follows the same lines as the CLT proof in [5] and exploits well-known results of the Stein-Malliavin method detailed in [26], in particular Proposition 3.1 (Theorem 5.1.3 [26]). To this purpose we will need some technical lemmas which can be found in Appendix 6.4. Fundamental is the computation of the mean and the variance. The latter was computed in the previous section where we showed that each chaotic component of the boundary length has finite asymptotic variance. However, the first step toward the CLT is the evaluation of the mean. In the following proposition we obtain the exact expression of E[L j (z)].
Proof. EL j (z) is the limit of EL ε j (z) for ε → 0, i.e. the limit of As showed in section 2,β j (x),∂ 1βj (x),∂ 2βj (x) are all standard normal random variables for every x ∈ S 2 . Therefore, Then we obtain Let us denote nowL j (z) := L j (z) − E[L j (z)]. As mentioned above, before proving our main result we need some technical lemmas (see Appendix 6.4 for the proofs). Let us first introduce the notation Lemma 5.2. For integers q, q the following bounds hold: where the constant C z depends only on the level z.
Keeping in mind the results for the variance obtained in the previous section, we prove the Central Limit Theorem following the same steps as in Theorem 9 in [5].
Proof of Theorem 1.1. Let us start by introducing the following notatioñ By triangle inequality we have For the first part, since the Wasserstein distance can be bounded by the L 2 norm, we get with the notation from Lemma 4.2. Now we note that the covariance functions in the integrand are 1 only at the origin, hence, we split the interval of the integral in [0, C/ ] and [C/ , π]. Following the same argument as in [19] and [8] we can write that, in [C/ , π], up to a constant we have for some δ > 0. Note that due to the covariance structure of the terms involved in the integrand (see (2.2)) δ can be chosen to be close to one (this can be seen by means of Hilb's asymptotics and similar approximations, see Proposition 6.4). Now The last step is due to Cauchy-Schwarz inequality applied as in [8]. The map (a, b, c) → Moreover, we have the estimate (2a+2b+c)! (2a)!(2b)!c! ≤ 3 2a+2b+c . Now we consider the sums a,b,c≥0 2a+2b+c≥N x 2a+2b+c for some 0 < x < 1. We have that a,b,c≥0 2a+2b+c≥N The first summand is (up to a constant independent of N ) of order N 2 x N −1 , the second one of order N x N −1 and the last one is of order x N −1 . In total, the sum is of order N 2 x N −1 . Applying this to the sums in (5.4) for x = 3 √ 1 − δ and √ 1 − δ respectively we obtain the bound (up to a constant) A 2 j N 4 (3(1 − δ)) N . Now for θ ∈ [0, C/ ], in light of Lemma 4.4, we have that converges and hence the tail of the series (q = N + 1, . . . , ∞) goes to zero.
For the second summand on the right hand side of (5.3) we procede similarly to [5]. By Proposition 3.1 we have We know that the denominator is asymptotically constant in both j and N . Moreover, since the Wiener chaos decomposition is orthogonal, we obtain Therefore, we have where the last step follows by Theorem 2.9.1 in [26] applied to the first summand and Cauchy-Schwarz inequality for the second one. By Remark 5.3 we can now bound this by Now we can use the result (5.2) and bound this further by which for big j and M > 4 is Finally for the third summand in (5.3) by Proposition 3.6.1 in [26], we have that which, as shown in the first part of the proof, is bounded by C 2 A 2 2 j [N 4 (3(1 − δ)) N ] 2 . Now, we put together the bounds we obtained for the three terms of the right hand side of (5.3) and choosing N := N (j) = j, δ > 11 12 and M > 4 we have that all the three terms go to zero as j → ∞ and the thesis of the theorem follows.

Properties of needlet systems
In this section we recall some analytic properties of the needlet systems, in particular their localization properties in real and harmonic spaces (see Theorem 3.5 [24]). The latter allows to show that needlet coefficients are asymptotically uncorrelated for any fixed angular distance, as the frequency goes to infinity. Let us consider the kernel In Theorem 13.1 [16] (see also [3] and Proposition 10.5 [16]) the authors showed that for all M ∈ N there exists a positive constant C M such that Moreover, since it is known that the derivative of an operator kernel related to the Laplace-Beltrami operator on the sphere is a kernel itself (see equation (12) in [24]) we can establish the same property on the kernels ∂ ∂ x,y Ψ (1) j ( x, y ) applying Lemma 2.1 [29] (see also Corollary 5.3 [24]). Hence, we get where C M , C M are positive constants.
Proof. The proof of all inequalities follows the same steps as the one in Lemma 10.8 [16]; where (6.7) has been proved. In order to get (6.8) we have to bound Note that condition 2.1 implies that c 1 −α ≤ C ≤ c 2 −α , then we have and Now we define the sequence η j := b 2 (x)x −α g j (x). As pointed out in [16], η j satisfies the same conditions as b(·) which ensures the localization property. Hence (6.4) implies that This result together with (6.12) and (6.11) gives

Other auxiliary results
A technical result used to simplify expectations of products of Hermite polynomials is the so called diagram formula (see [16] for details). Let us introduce the necessary notation.
For an integer p ≥ 1 and ( 1 , . . . , p ) ∈ N p a diagram γ of order ( 1 , . . . , p ) is a graph of 1 vertices indexed by 1, 2 vertices indexed by 2 and so forth until p (this can be viewed as vertices ordered in p rows) with each vertex having degree 1. We denote by η ik (γ) the number of edges connecting vertices in rows i and k for a given diagram γ. We say that a diagram has no flat edges if there are no edges connecting vertices belonging to the same row. We denote by Γ( 1 , . . . , p ) the set of all diagrams of order ( 1 , . . . , p ) with no flat edges.
We can now formulate the diagram formula for moments.
It follows that The identities now follow by orthogonality of Legendre polynomials.

Proofs of Lemma 4.4 and Proposition 4.5
Proof of Lemma 4.4. Let us introduce the following notation: .., is the appropriately scaled real basis of the th eigenspace (see [16] for the precise relation between Y * m and Y m and the respective coefficients a * m and a m ) such that Let n be the number of components of the vectors defined above.
Recalling that with the new notation we can write the integrand above as Write, moreover K(x, y) := lim ε1, ε2→0 K ε1ε2 (x, y). Note that by isotropy we have where N = (0, 0) and x = (θ, 0), and the integral that we want to study is With the new definitions given above we have Let us write using the definition of dνβ K ε1ε2 (x, y) = 1 4ε 1 ε 2 We have:β j (x) = U (x),β j = cos α U (x),βj U (x) β j , which is, up to the sign, equal to the length of the projection ofβ j on the line generated by U (x). Let us denote by π the plain spanned by U (x) and U (y). It follows that the above domain of integration are vectors whose projections onto π lie in one of the red parallelograms in Figure 1 (note that for z = 0 these parallelograms coincide). Each of these parallelograms is of the size of the parallelogram P in Lemma 3.4 in [33], namely 4ε 1 ε 2 1 √ 1−u(x,y) 2 , where u(x, y) is the quantity defined above (and different from u(x, y) in [33]). We call the figure comprised of the four parallelograms P z and rewrite as in [33]:

Writing out the inner products for
such that the above integral can be written as Note that for f 1 ∈ π ⊥ we have by triangle inequality for the norms induced by U(x) and U(y) since p T U(x)p is bounded by tr(U(x)). Thus, under an appropriate rotation O, the inner integral can be bounded up to an absolute constant by for X = (0, 0, X 3 , . . . , X n ) T , where X 3 , . . . , X n are standard normal independent random variables. By the Cauchy-Schwarz inequality we obtain 1). Recalling that by properties of the trace and due to the fact that has nonnegative diagonal entries, we have and thus, Therefore, we arrive at the bound Similarly, we can see that In total, we can bound the inner integral by A 2 j O(1). We obtain thus Consequently, we have Note that with our above definitions we have Therefore, the integral for small θ is bounded by We have From the Taylor approximation P (cos θ) = 1 − θ 2 ( +1) 2 + θ 4 O( 4 ) we conclude 1 − P (cos θ)P (cos θ) and thus, It follows that Proof of Proposition 4.5. We have We need to understand the asymptotics of the terms A 2 j I (u1, k1, u2, k2) q, j . First note that due to isotropy of f we can reparametrise the integrals as where x = (0, 0) is the north pole of the sphere S 2 and y = (θ, 0). By Lemma 4.2 (and with its notation) this integral (divided by 8π 2 ) becomes (cos θ)(cos θ) γ1 P α+β+γ 1 +1 (cos θ) . . . P α+β+γ (cos θ)(sin θ) 2(γ−γ1) ×P α+β+γ (cos θ) . . . P q (cos θ) sin θdθ αβγ .
Note that the functions P n (cos(· )), P n (cos(· )), P n (cos(· )), cos(· ) and sin(· ) are either line-or point-symmetric at π 2 , and therefore the integrals for each of the constellations will be either zero or twice the integrals between zero and π 2 . One can see from (2.2) and the identity P n (cos θ) cos θ − P n (cos θ) sin 2 θ = (n + 1) 2 P n (cos θ) − P n+1 (cos θ) that the integrals cancel if and only if q i=1 i is odd. Therefore, for the asymptotics it suffices to consider integrals ranging from zero to π 2 . Moreover, let us substitute θ by ψ , where = 2 j . We obtain for ( 1 . . . , q ) such that the integrand is line-symmetric where the two terms IntS αβγ represent the integral ranging from zero to ε and from ε to π/2 respectively. For the integrals away from zero we can use the asymptotics given in Proposition 6.4 and expand the sum in the expansion of the second derivatives. Denoting by ∼ asymptotic equivalence for large , we can write with γ 2 = γ − γ 1 and γ 2 = γ 2 − γ 2 . Now use the asymptotics sin(x) ≈ x as well as cos(x) ≈ 1 and where ∼ means asymptotic equivalence up to a constant with respect to and For q > 4 and q = 3 the integral in which all the factors of IntL (γ1, γ 2 ) αβγ depending on ψ are collected, is finite (see Lemma 6.5 for the proof). Therefore, we have i . Now recall that using the asymptotics given in Remark 2.3 one can write Performing a similar calculation for other i (i = 2, . . . , q), we arrive at and consequently, using the asymptotic results for A 2 j and B 2 j , For IntS we note that the asymptotic formula from [30] for θ ∈ (0, ε/ ) P n (cos θ) = θ sin θ 1/2 J 0 ((n + 1/2)θ) + δ n (θ) with δ n (θ) θ 2 ≤ ε 2 2 and a Bessel function J 0 can be transferred via classical identities (see [1]) and P n (x) = 1 1 − x 2 (2xP n (x) − n(n + 1)P n (x)) to P n (cos θ) = n sin 2 θ θ sin θ 1/2 J 0 ((n − 1/2)θ) + δ n−1 (θ) − cos θ θ sin θ 1/2 J 0 ((n + 1/2)θ) − cos θδ n (θ) and P n (cos θ) cos θ − P n (cos θ) sin 2 θ = n(n + 1)P n (cos θ) − P n (cos θ) cos θ = n(n + 1) θ sin θ 1/2 J 0 ((n + 1/2)θ) + n(n + 1)δ n (θ) After the substitution θ = ψ the terms that one then needs to consider in order to establish convergence are of the form and for these terms we have similarly to Remark 2.3 Note that the contribution of terms containing δ n is smaller than that of the others and can be ignored. Moreover, we have by normalisation, i.e. the integrals close to zero are bounded. Therefore, the order with respect to 2 j resulting from the evaluation of terms in (6.13) is at most constant and the limit with respect to j is well-defined. This finishes the proof.
Lemma 6.5. With the notation from Proposition 4.5 set ζ := α + γ 2 . Then for q = 3, q > 4, and q = 4 ζ = 1, 3 (6.14) we have that where Proof. It is easily seen that the constant C ψ can be bounded by 1 and then if α+β+γ 2 − as → ∞. For ζ = 0 we can see in the same way that For ζ = 1, 2 we obtain, respectively, which converge, as we have just shown.

On the proof of CLT
In this section we give the proofs of the technical lemmas we used to prove the Central Limit Theorem.
Each integral in this expression is a double sum over permutations of q and q divided by q!q ! of double integrals dσ(x 1 )dσ(x 2 ) emerging from the definition of g qjuk . The entries at the last r positions integrate to covariances of the form E[f 1 (x 1 )f 2 (x 2 )] with f 1 and f 2 being eitherβ j ,∂ 1βj or∂ 2βj . We can see that g q,j ⊗ rgq ,j 2 = A 2 2 j q u1=0 u1 k1=0 q u 1 =0 where F 1 , F 2 , F 3 and F 4 are products of respectively q − r, q − r, r and r of the covariances involvingβ j ,∂ 1βj and∂ 2βj . The specific terms depend on the choice of permutations σ, σ , σ 1 and σ 1 . x, y positive, we can bound

Now we note that
and then with some constant K M depending only on M . We conclude that for big j, using the asymptotics of A 2 j . The second bound in (5.1) is obtained in a similar way from the inequality (2q − 2r)! g q,j (y 1 , . . . , y q ) ⊗ rgq,j (y 1 , . . . , y q ) 2 H 2q−2r .
Proof of Lemma 5.4. Let us consider the sum u k=0 α k,u−k k!(u − k)! .
By definition α k,u−k is zero except for k, u even. Hence we set k = 2k and u = 2u to write with some constant C z . Let us denote q := q/2 and estimate this term by This last expression can be estimated by C z 1 q ! q u=0 q u which equals C z 1 q ! 2 q . Note that for large n we have 1 n! ∼ 2 n n −1/4 1 √ (2n)! by Stirling formula, and hence we obtain the bound 1 q ! 2 q ≤ 2 q 1 (q − 1)! which leads to (5.2).