aa r X i v : . [ m a t h . S T ] M a y Electronic Journal of Statistics
Vol. 2 (2008) 332–367ISSN: 1935-7524DOI:
The needlets bispectrum ∗ Xiaohong Lan
Institute of Mathematics, Chinese Academy of Sciences andDepartment of Mathematics, University of Rome Tor Vergatae-mail: [email protected]
Domenico Marinucci
Department of Mathematics, University of Rome Tor Vergatae-mail: [email protected]
Abstract:
The purpose of this paper is to join two different threads ofthe recent literature on random fields on the sphere, namely the statisti-cal analysis of higher order angular power spectra on one hand, and theconstruction of second-generation wavelets on the sphere on the other. Tothis aim, we introduce the needlets bispectrum and we derive a number ofconvergence results. Here, the limit theory is developed in the high resolu-tion sense. The leading motivation of these results is the need for statisticalprocedures for searching non-Gaussianity in Cosmic Microwave Backgroundradiation.
AMS 2000 subject classifications:
Primary 62G20; secondary 62M15,60B15, 60G60.
Keywords and phrases:
Bispectrum, Needlets, Spherical Random Fields,Cosmic Microwave Background Radiation, High Resolution Asymptotics.Received February 2008.
Contents ∗ We are grateful to a Referee and an Associate Editor for a very careful reading andmany insightful remarks on an earlier version of this paper. We are also grateful to P.Baldi,G.Kerkyacharian and D.Picard for many useful discussions. The research of Xiaohong Lanwas supported by the Equal Opportunity Committee of the University of Rome Tor Vergata.332 . Lan and D. Marinucci/Needlets bispectrum
1. Introduction
The purpose of this paper is to join two different threads of the recent literatureon random fields on the sphere, namely the statistical analysis of higher orderangular power spectra on one hand, and the construction of second-generationwavelets on the sphere on the other hand. More precisely, we shall be concernedwith zero-mean, mean square continuous and isotropic random fields on thesphere, for which the following spectral representation holds [1, 4]: for x ∈ S ,T ( x ) = X l,m a lm Y lm ( x ) (1)where { a lm } l,m , m = 1 , . . . , l is a triangular array of zero-mean, orthogonal,complex-valued random variables with variance E | a lm | = C l , the angular powerspectrum of the random field. For m < a lm = ( − m a l − m , whereas a l is real with the same mean and variance. (1) holds in the L ( S ) sense, i.e.we have lim L →∞ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S ( T ( x ) − L X l =1 l X m = − l a lm Y lm ( x ) ) µ ( dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0, µ ( dx ) denoting the uniform measure on the sphere. The functions { Y lm ( x ) } arethe so-called spherical harmonics, i.e. the eigenvectors of the Laplacian operatoron the sphere,∆ S Y lm ( ϑ, ϕ ) = (cid:20) ϑ ∂∂ϑ (cid:26) sin ϑ ∂∂ϑ (cid:27) + 1sin ϑ ∂ ∂ϕ (cid:21) Y lm ( ϑ, ϕ )= − l ( l + 1) Y lm ( ϑ, ϕ )where we have moved to spherical coordinates x = ( ϑ, ϕ ), 0 ≤ ϑ ≤ π and0 ≤ ϕ < π . It is a well-known result that the spherical harmonics provide acomplete orthonormal systems for L ( S ) [36] . The analysis of random fields on the sphere has recently gained very strongphysical motivations, due to the overwhelming amount of data which is becom-ing available on Cosmic Microwave Background radiation (hereafter (CMB)).As detailed elsewhere [15], to the first order we can view CMB data as mapsof the Universe in the immediate adjacency of the Big Bang. The first of thesemaps were provided by the satellite experiment
COBE in 1993, and in viewof this G. Smoot and J. Mather were awarded the Nobel prize for Physics in2006. Much more refined maps have been made available by another NASAsatellite experiments,
WMAP ( http://wmap.gsfc.nasa.gov/ ); still more re-fined data are expected from the ESA mission Planck, scheduled to be launchedin October 2008. These huge collaborations involve hundred of scientists andare expected to provide invaluable information on Physics and Cosmology. Atthe same time, these massive data sets have called for huge statistical chal-lenges, ranging from power spectrum estimation to outlier detection, testing . Lan and D. Marinucci/Needlets bispectrum for isotropy, efficient denoising and map-making, handling with missing data,testing for non-Gaussianity and many others.Among these issues, particular interest has been driven by efficient testingfor non-Gaussianity. This is due to strong physical motivations (the leadingparadigm for the Big Bang dynamics predicts (very close to) Gaussian fluctua-tions) and difficulties in finding a proper statistical procedure. There is by now awide consensus that the most efficient procedures to probe non-Gaussianity arebased upon the bispectrum, in the idealistic circumstances where the sphericalrandom field is fully observed: see for instance [8, 11, 20, 23, 24, 37]. However, theproperties of the bispectrum are also known to deteriorate dramatically in thepresence of missing observations, see again [11]. To handle the latter problem,a general approach is to focus on wavelet, rather than Fourier transforms. Theconstruction of spherical wavelets has recently drawn an enormous amount ofattention in the literature, see for instance [2, 27, 38] and the references therein.In our view, a particularly convenient tight frame construction on the sphere isprovided by so-called needlets, which were introduced in [30, 31]; their applica-tions to spherical random fields is due to [6, 7]. Needlets enjoy two propertieswhich seem especially worth recalling: they are quasi-exponentially localized inthe real domain and compactly supported in the harmonic domain. A further,quite unexpected property is as follows: random needlets coefficients are asymp-totically uncorrelated at the highest frequencies and hence, in the Gaussian case,independent, see again [6]. This latter feature is rather surprising in a compactdomain and makes asymptotic theory possible even in the presence of a singlerealization of a spherical random field.Our aim in this paper is to borrow ideas from the bispectrum and the needletsliterature to propose and analyze a needlets bispectrum, where the random co-efficients in the needlets expansion are combined in a similar way to the bispec-trum construction. The aim is to obtain a procedure which mimicks the abilityof the bispectrum to search for non-Gaussianity at the most efficient combina-tion of frequencies, at the same time providing a much more robust constructionin the presence of missing data, as typical of the needlets. The plan of the pa-per is as follows: in Section 2 we review some background material on sphericalrandom fields, the bispectrum and the needlets construction. In Section 3 weintroduce the needlets bispectrum and we establish a central limit theorem, inthe high resolution sense. In Section 4 we go on to establish a functional centrallimit theorem for the integrated needlets bispectrum; in Section 5 we providesome preliminary discussion on the behaviour under non-Gaussian assumptionsand discuss some possibilities for applications and further research.
2. Some background material
In this subsection we shall briefly recall the main features of the needlets con-struction. As mentioned above, needlets were first introduced in the Functional . Lan and D. Marinucci/Needlets bispectrum
Analysis literature by [30, 31], whereas the investigation of their properties froma stochastic point of view is due to [5, 6] and [7]; see also [19, 17]. We need firstto introduce some notation and definitions, which are largely identical to thosein [6, 31].Given any two positive sequences { a j } , { b j } , we write a j ≈ b j if there exist c > c − a j ≤ b j ≤ ca j for all j. The standard (open and closed)balls in S are given as always by B ( a, α ) = { x, d ( a, x ) ≤ α } , B ◦ ( a, α ) = { x, d ( a, x ) < α } . For a general A ⊂ S we will denote by | A | the sphericalmeasure of A . Now fix ǫ > x , . . . x N in S such that ∀ i = j, d ( x i , x j ) >ǫ ; the set { x , . . . , x N } = Ξ ǫ is called a maximal ǫ − net if it satisfies ∀ x ∈ S , d ( x, Ξ ǫ ) ≤ ǫ , ∪ x i ∈ Ξ ǫ B ( x i , ǫ ) = S and ∀ i = j, B ( x i , ǫ/ ∩ B ( x j , ǫ/
2) = ∅ .It follows from Lemma 5 in [7] that4 ǫ ≤ N ≤ ǫ π (2)For all x i ∈ Ξ ǫ , the associated family of Voronoi cells is defined by: V ( x i ) = { x ∈ S , ∀ j = i, d ( x, x i ) ≤ d ( x, x j ) } .We recall that B ( x i , ǫ/ ⊂ V ( x i ) ⊂ B ( x i , ǫ ). Also, if two Voronoi cells areadjacent, i.e. V ( x i ) ∩ V ( x j ) = ∅ , then by necessity d ( x i , x j ) ≤ ǫ . It is proved in[7] that there are at most 6 π adjacent cells to any given cell.For the construction of needlets, we should first start to define K l as the spaceof the restrictions to the sphere S of polynomials of degree less than l . The nextingredient are the set of cubature points and cubature weights; indeed, it is nowa standard result (see for instance [29]) that for all j ∈ N , there exists a finitesubset X j of S and positive real numbers λ jk >
0, indexed by the elements of X j , such that ∀ f ∈ K l , Z S f ( x ) dx = X ξ jk ∈X j λ jk f ( ξ jk ). (3)Given a fixed B > , we shall denote by { ξ jk } the cubature points correspondingto the space K [3 B j +1 ] , where [ . ] represents as usual integer part. It is known that {X j } ∞ j =0 can be taken s.t. the cubature points for each j are almost uniformly ǫ j − distributed with ǫ j := κB − j , and the coefficients { λ jk } are such that λ jk ≈ cB − j , card {X j } ≈ B j .Now let φ be a C ∞ function supported in | ξ | ≤
1, such that 0 ≤ φ ( ξ ) ≤ φ ( ξ ) = 1 if | ξ | ≤ B − , B >
1. Following again [30, 31], we define b ( ξ ) = φ (cid:18) ξB (cid:19) − φ ( ξ ) ≥ ∀| ξ | ≥ , X j b (cid:18) ξB j (cid:19) = 1. (4)It is immediate to verify that b ( ξ ) = 0 only if B ≤ | ξ | ≤ B . The needlets frame { ϕ jk } is then constructed as ϕ jk ( x ) := p λ jk X l b (cid:18) lB j (cid:19) l X m = − l Y lm ( ξ jk ) Y ∗ lm ( x ). (5) . Lan and D. Marinucci/Needlets bispectrum The main localization property of { ϕ jk } is established in [30], where it is shownthat for any M ∈ N there exists a constant c M > ξ ∈ S : | ϕ jk ( ξ ) | ≤ c M B j (1 + B j arccos h ξ jk , ξ i ) M uniformly in ( j, k ) . More explicitly, needlets are almost exponentially localized around any cubaturepoint, which motivates their name. In the stochastic case, the (random) sphericalneedlet coefficients are then defined as β jk = Z S T ( x ) ϕ jk ( x ) dx = p λ jk X l b (cid:18) lB j (cid:19) l X m = − l a lm Y lm ( ξ jk ) . (6)We have immediately X k β jk p λ jk = 0 , (7)i.e., the (weighted) sample mean of the needlets coefficients is identically zeroat all levels j. The proof is trivial, because X k β jk p λ jk = B j +1 X l = B j − l X m = − l b (cid:18) lB j (cid:19) a lm "X k λ jk Y lm ( ξ jk ) = B j +1 X l = B j − l X m = − l b (cid:18) lB j (cid:19) a lm (cid:20)Z S Y lm ( x ) dx (cid:21) = 0.The variance of the needlets coefficients is given by E β jk = λ jk B j +1 X l = B j − b (cid:18) lB j (cid:19) C l l X m = − l Y lm ( ξ jk ) Y lm ( ξ jk )= λ jk B j +1 X l = B j − b (cid:18) lB j (cid:19) C l l + 14 π P l (cos 0)= λ jk B j +1 X l = B j − l + 14 π b (cid:18) lB j (cid:19) C l =: σ jk > σ jk ≈ : σ j uniformly over k, where σ j := 4 πN j B j +1 X l = B j − l + 14 π b (cid:18) lB j (cid:19) C l , N j = card {X j } . From now on, we shall typically focus on the normalized needlets coefficients,defined as b β jk := β jk /σ j . To investigate the correlation, we introduce now the same, mild regularityconditions on the angular power spectrum C l of the random field T ( x ) as in[5, 6]. . Lan and D. Marinucci/Needlets bispectrum Condition A
The random field T ( x ) is Gaussian and isotropic with angularpower spectrum such that, for all B > , there exist α > , and { g j ( . ) } j =1 , ,... a sequence of functions such that C l = l − α g j (cid:18) lB j (cid:19) >
0, for B j − < l < B j +1 , j = 1 , , . . . (8)where c − ≤ g j ≤ c for all j ∈ N , andsup j sup B − ≤ u ≤ B (cid:12)(cid:12)(cid:12)(cid:12) d r du r g j ( u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c r some c , c , . . . c M > , M ∈ N .Condition 8 entails a weak smoothness requirement on the behaviour of theangular power spectrum, which is trivially satisfied by some cosmologically rel-evant models (where the angular power spectrum usually behaves as an inversepolynomial, see again [15], pp. 243–244). For instance, considering positive con-stants d j , j = 1 , . . . , p, p > , for all B > C l = 1 P pk =0 d k l k = l − p l p d + d l + ...d p l p = l − p ( l/B j ) p P pk =0 d k B jk − jp ( l/B j ) k = l − p g j (cid:18) lB j (cid:19) , for B j − < l < B j +1 , j = 1 , , . . .g j ( u ) := u p P pk =0 d k B jk − jp u k .Under Condition 8 a crucial and rather unexpected property of the randomneedlets coefficients is established in [6], namely the correlation bound | Corr ( β jk , β jk ′ ) | ≤ c M (1 + B j d ( ξ jk , ξ jk ′ )) M (9)where d ( ξ jk , ξ jk ′ ) = arccos( h ξ jk , ξ jk ′ i ). In words, (9) is stating that as the fre-quency increases, spherical needlets coefficients are asymptotically uncorrelated(and hence, in the Gaussian case, independent), for any given angular distance.This property is of course of the greatest importance when investigating theasymptotic behaviour of statistical procedures: in some sense, it states that itis possible to derive an infinitely growing array of asymptotically independent“observations” (the needlets coefficients) out of a single realization of a continu-ous random field on a compact domain. It should be stressed that this propertyis not by any means a consequence of the localization properties of the needletsframe. As a counterexample, it is easy to construct spherical frames havingbounded support in real space, whereas the corresponding random coefficients . Lan and D. Marinucci/Needlets bispectrum are not at all uncorrelated (recall the angular correlation function can be takento be bounded from below at any distance on the sphere).We recall briefly two further results that we shall exploit often in the sequel;more precisely, ([6], Lemma 10 ) we note that for
M > , j ∈ N , ∀ k, k ′ , X ξ jk ∈ χ j B j d ( ξ jk , ξ jk ′ )) M ≤ C M some C M >
0. (10)Also ([31],
Lemma 4.8 ), for some C ′ M depending only on M we have the in-equality X ξ jk ∈ χ j B j d ( ξ jk , ξ jk ′ )) M B j d ( ξ jk , ξ jk ′′ )) M ≤ C ′ M (1 + B j d ( ξ jk ′ , ξ jk ′′ )) M .(11)From the computational point of view, we should stress that needlets arenot only feasible, but indeed extremely convenient. The implementation canbe performed with a minimal effort by means of standard packages for theanalysis of spherical random fields such as HEALPIX or GLESP ([18] and [16]),see for details [26], where plots and numerical evidence on localization anduncorrelation are also provided. To complete our background, we need a quick review on the diagram formula.This is material which can now be found at a textbook level (see for instance[34]); nevertheless, we need a brief overview to fix notation. Denote by H q the q − th order Hermite polynomials, defined as H q ( u ) = ( − q e u / d q du e − u / . We now introduce diagrams , which are basically mnemonic devices for com-puting the moments and cumulants of polynomial forms in Gaussian randomvariables. Our notation is the same as for instance in [23, 24]. Let p and l j ,j = 1 , . . . , p be given integers. A diagram γ of order ( l , . . . , l p ) is a set of points { ( j, l ) : 1 ≤ j ≤ p, ≤ l ≤ l j } called vertices, viewed as a table W = −→ l ⊗· · ·⊗−→ l p and a partition of these points into pairs { (( j, l ) , ( k, s )) : 1 ≤ j ≤ k ≤ p ; 1 ≤ l ≤ l j , ≤ s ≤ l k } , called edges. We denote by Γ( W ) the set of diagrams of order ( l , . . . , l p ). If theorder is l = · · · = l p = q , for simplicity, we also write Γ( p, q ) instead of Γ( W ).We say that: a ) A diagram has a flat edge if there is at least one pair { ( i, j )( i ′ , j ′ ) } suchthat i = i ′ ; we write Γ F for the set of diagrams that has at least one flat edge,and Γ F otherwise. . Lan and D. Marinucci/Needlets bispectrum b ) A diagram is connected if it is not possible to partition the rows −→ l · · · −→ l p ofthe table W into two parts, i.e. one cannot find a partition K ∪ K = { , . . . , p } that, for each member V k of the set of edges ( V , . . . , V r ) in a diagram γ , either V k ∈ ∪ j ∈ K −→ l j , or V k ∈ ∪ j ∈ K −→ l j holds; we write Γ C for connected diagrams, andΓ C otherwise. c ) A diagram is paired if, considering any two sets of edges { ( i , j )( i , j ) }{ ( i , j )( i , j ) } , then i = i implies i = i ; in words, the rows are completelycoupled two by two. We write Γ P for the set of diagrams for paired diagrams,and Γ P otherwise. Proposition 1. (Diagram Formula)
Let ( z , . . . , z p ) be a centered Gaussianvector, and let γ ij = E [ z i z j ] , i, j = 1 , . . . , p their covariances, Let H l , . . . , H l p beHermite polynomials of degree l , . . . , l p respectively. Let L be a table consistingof p rows l , . . . l p , where l j is the order of Hermite polynomial in the variable z j . Then E [Π pj =1 H l j ( z j )] = X G ∈ Γ( l ,...,l p ) Π ≤ i ≤ j ≤ p γ η ij ( G ) ij Cum ( H l ( z ) , . . . , H l p ( z p )) = X G ∈ Γ c ( l ,...,l p ) Π ≤ i ≤ j ≤ p γ η ij ( G ) ij where, for each diagram G , η ij ( G ) is the number of edges between rows l i , l j and Cum ( H l ( z ) , . . . , H l p ( z p )) denotes the p − th order cumulant. We have now all the preliminary material to define our needlets bispectrumon S , as explained in the following Section.
3. A Central Limit Theorem for the Needlets Bispectrum
As mentioned in the introduction, the recent literature suggests that the mostpowerful statistic to search for non-Gaussianity in fully observed spherical ran-dom fields is the (normalized) angular bispectrum, defined as I l l l = X m m m (cid:18) l l l m m m (cid:19) a l m a l m a l m p C l C l C l where the symbol in brackets represents the so-called Wigner’s 3 j coefficients,which are meant to ensure the statistics is rotationally invariant. Wigner’s 3 j coefficients arise in many different instances, especially in the quantum theoryof angular momentum (see [36], where explicit expressions are also provided).Up to a normalization factors, they are equivalent to the so-called Clebsch-Gordan coefficients, which play an important role in representation theory forthe group of rotations SO (3), see [25] for a much more detailed discussion and . Lan and D. Marinucci/Needlets bispectrum probabilistic applications. Following [20, 37], an alternative definition of the(normalized) bispectrum can be considered, namely e I l l l = Z S T l ( x ) T l ( x ) T l ( x ) p V ar ( T l ( x )) V ar ( T l ( x )) V ar ( T l ( x )) dx (12)where T l ( x ) := L l ∗ T = Z S T ( y ) X m Y lm ( x ) Y ∗ lm ( y ) dy = X m a lm Y lm ( x ),i.e. we focus on the Fourier projections of the random fields at multipoles( l , l , l ) . Both versions of the bispectrum have been shown to be extremelypowerful against non-Gaussian alternatives, indeed there is now a widespreadconsensus that they make up the most efficient statistical procedures to searchfor non-Gaussianity, at least in the presence of fully observed spherical maps(see for instance [11, 21, 28]).On the other hand, it is also well-known that the performance of Fouriermethods in general, and the bispectrum in particular, deteriorates quite clearlyin the presence of missing observations/partially observed maps ([20, 22, 11]).A natural idea is then to explore the localization properties of the needletsin harmonic domain, together with their robustness in the presence of gaps,in order to devise a statistic which might mimick the positive features of thebispectrum, at the same time coping with the difficulties brought in by missingobservations.To this aim, we shall consider the (normalized) needlets bispectrum I j j j := X k b β j k b β j k b β j k δ j j j ( k , k , k ) h j j j ( k , k , k ), j ≤ j ≤ j ,(13)where δ j j j ( k , k , k ) = I ( ξ j k ∈ V j k ) I ( ξ j k ∈ V j k ), h j j j ( k , k , k ) = B j − j √ λ j k { k : k ∈V j k ∩X j } , j < j = j p λ j k , otherwise , where I ( . ) denotes the indicator function and V jk is the Voronoi Cell that corre-sponds to ξ jk ; note that for j = j we have h j j j ≈ p λ j k . It is immediate tosee that (13) can be seen as a natural development of (12), where the convolu-tion with the orthonormal projector operator L l is replaced by the (discretized)convolution with the frame operator projection p λ jk P l b ( l/B j ) L l . Of course,in practice (12) is unfeasible and requires discretization to be implemented. Inwords, we are considering a version of the bispectrum where the exact identi-fication of the multipoles is blurred by a form of suitable smoothing, with thepurpose of a better robustness against missing observations.The summation convention in (13) needs some further discussion. In practice,for applications the Voronoi tessellation is chosen in such a way to be nested . Lan and D. Marinucci/Needlets bispectrum across different scales (this is the case, for instance, for HEALPix [18], whichis the standard package for CMB applications). Under such circumstances, ourprocedure can be described more explicitly as follows: we take I j j j := X k b β j k b β j k b β j k h j j j ( k , k , k ), (14)where k = k ( k ) is the (unique) value of k such that ξ j k ∈ V j k , and k = k ( k ) is the (unique) value of k such that ξ j k , ξ j k ∈ V j k . In otherwords, the “finest grid” X j is the one which leads the summation, whereassmaller frequency terms are identified with those centres whose correspondingVoronoi cells include the points being summed. Note, however, that in the gen-eral non-nested case the centre of V j k needs not belong to X j . In the sequel,for notational simplicity we write k , k rather than k ( k ) , k ( k ) , when noambiguity is possible.To investigate the asymptotic behaviour of the needlets bispectrum, we shallmake an extensive use of the Diagram Formula which was introduced in theprevious section. A crucial point, of course is the determination of the frequencieswhere the needlets bispectrum is evaluated. We distinguish three cases, namely: • Case 1) j + 1 < j < j − • Case 2) j + 1 < j = j , or j = j < j − • Case 3) j = j = j . Case 1 corresponds to the situation where all three frequencies are different.Case 2) is basically the “squeezed” or collapsed configuration which is consid-ered by [3, 23], and many other cosmological references; in words, one frequencyis (much) smaller than the other two. It has been widely argued in the physicalliterature that this configuration corresponds to the highest power region forso-called local models of non-Gaussianity. Case 3 corresponds to so-called equi-lateral configurations; this case, however, can be largely investigated by meansof results in [6] and we report it only for completeness, omitting many detailsin the proof. It should be noted that for case 1) and 2) we focus on frequenciesthat are at least two steps apart, in order to exploit the semiorthogonality prop-erties of the needlets systems. Relaxing this assumption implies no new ideasand would only make the paper notationally more complicated.In each of the three cases we have trivially E I j j j = 0 . We now focus onthe asymptotic behaviour; here, asymptotics must be understood in the highresolution sense, i.e. we focus on a single realization of an isotropic random field,and we investigate the behaviour of our statistics at higher and higher bands.The first task is to ensure the statistics are non-degenerate and do not exhibitan explosive behaviour; this is the aim of the next Lemma. While the boundfrom above is quite straightforward, the lower bound is much more complicatedand settles a question which was left open in [5], where the lower limit wassimply assumed to be strictly larger than zero even for the simple case where j = j = j . As it is evident from the proof, the integer K depends on thechoice of cubature points and of kernel function b ( . ); more explicit expressions . Lan and D. Marinucci/Needlets bispectrum can be found below. Note also that unless the three bands are equal, conditionb) cannot be satisfied for B = 2; indeed in CMB applications values of order B ≃ . , . Lemma 1.
Under Condition A, as j → ∞ , a) For all j ≤ j ≤ j , E I j j j = O (1) .Also, there exist a positive integer K such thatb) For { j = j = j } , or (cid:8) j + K < j , B j + B j ≥ B j (cid:9) , (cid:8) E I j j j (cid:9) − = O (1) . The bounds are uniform with respect to j , j . Proof. (a)
In the sequel, we shall use the fact that the set of the cubature pointsof polynomial spaces with degree less than B j are a κB − j − net; we also define ρ := max j,k { B − j λ j,k } . The proof of all three cases are similar; we shall focuson case 2) j + 1 < j = j . E I j j j = X k ,k ′ ∈X j E β j k β j k ′ E ( β j k β j k ′ )( σ j σ j σ j ) h j j j ( k , k , k ) h j j j ( k ′ , k ′ , k ′ )= B j − j X k ,k ′ ∈X j X k ∈V j k ∩X j ,k ′ ∈V j k ′ ∩X j (cid:16) E b β j k b β j k ′ + 2 E b β j k b β j k ′ ( E b β j k b β j k ′ ) (cid:17) × p λ j k { k , k ∈ V j k ∩ X j } p λ j k ′ { k ′ , k ′ ∈ V j k ′ ∩ X j } = B j − j X k k ′ E β j k β j k ′ ( E β j k β j k ′ ) ( σ j σ j σ j ) p λ j k { k , k ∈ V j k ∩ X j }× p λ j k ′ { k ′ , k ′ ∈ V j k ′ ∩ X j } .Since λ jk ≃ B − j for every ξ jk , and { k , k ∈ V j k ∩ X j } ≃ B j − j , thesum can be readily bounded by CB j X k k ′ E b β j k b β j k ′ (cid:16) E b β j k b β j k ′ (cid:17) ≤ C ′ B j X k k ′ (cid:16) E b β j k b β j k ′ (cid:17) = O (1) as j → ∞ ,in view of (10). This completes the proof of part a). (b) The proof of the lower bound on the variance is considerably more deli-cate. We recall the correlation of the needlets coefficient is provided by E b β jk b β jk ′ ≈ P B j +1 l = B j − b (cid:0) lB j (cid:1) C l P l (cos θ ) P B j +1 l = B j − b (cid:0) lB j (cid:1) C l l +14 π . Lan and D. Marinucci/Needlets bispectrum where θ = arccos h ξ jk , ξ jk ′ i . The idea of our argument is to replace the needletscoefficients in the coarsest grid X j by coefficients with the same resolution butevaluated over the pixels X j ∩ V j ( ξ j k ) . This will allow us to circumvent thefact that the cubature weights at the smaller frequencies stay constant over aVoronoi cell, whereas those corresponding to the highest j ’s may vary. We shallhence consider β ∗ jk ( ξ j ′ k ′ ) = p λ jk B j +1 X l = B j − l X m = − l b (cid:18) lB j (cid:19) a lm Y lm ( ξ j ′ k ′ ).For fixed ( j ′ , k ′ ) , j ′ < j, β ∗ jk ( ξ j ′ k ′ ) varies over the pixels in X j ∩ V j ′ ( ξ j ′ k ′ ) . Let us now recall a few properties of Legendre polynomials, that we shall useextensively soon (see [36] for details); we havesup θ ∈ [0 ,π ] P l (cos θ ) = P l (cos 0) = 1 , and sup θ ∈ [0 ,π ] (cid:12)(cid:12)(cid:12)(cid:12) ddθ P l (cos θ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ l. As a consequence, for any positive ǫ < , there exists a δ > , s.t. if 0 ≤ θ <δ ≤ ǫ/ (3 l ) , then, | P l (cos( θ + θ )) − P l (cos θ ) | ≤ lθ ≤ ǫ, because 0 ≤ cos( θ + θ ) − cos θ = 2 sin 2 θ + θ θ ≤ θ .Now fix an integer K such that K ≥ log B κ/ǫ ; if we let { ξ j,k } j,k be the cu-bature points of polynomial space with degree less than B j + K +1 , (note thatwe can assume all of these sets are κB − j − nets), then for any j > j , ξ j k ∈V j k , ξ j k ′ ∈ V j k ′ , (cid:12)(cid:12)(cid:10) ξ j k , ξ j k ′ (cid:11) − (cid:10) ξ j k , ξ j k ′ (cid:11)(cid:12)(cid:12) = (cid:12)(cid:12) d ( ξ j k , ξ j k ′ ) − d (cid:0) ξ j k , ξ j k ′ (cid:1)(cid:12)(cid:12) ≤ κB − ( j + K +1) ≤ B − ( j +1) ǫ/ (cid:12)(cid:12)(cid:12) E β j k β j k ′ − E [ β ∗ j k ( ξ j k ) β ∗ j k ′ ( ξ j k ′ ) (cid:12)(cid:12)(cid:12) ≤ p λ jk λ jk ′ B j X l = B j − b (cid:18) lB j (cid:19) × C l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X m = − l Y lm ( ξ j k ) Y lm ( ξ j k ′ ) − l X m = − l Y lm ( ξ j k ) Y lm ( ξ j k ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = p λ jk λ jk ′ B j X l = B j − b (cid:18) lB j (cid:19) C l l + 14 π (cid:12)(cid:12)(cid:12) P l (cid:16)D ξ j k , ξ j k ′ E(cid:17) − P l (cid:16)D ξ j k , ξ j k ′ E(cid:17)(cid:12)(cid:12)(cid:12) ≤ ǫ p λ jk λ jk ′ B j X l = B j − b (cid:18) lB j (cid:19) C l l + 14 π ≤ ǫCσ j . (15) . Lan and D. Marinucci/Needlets bispectrum where C is a constant, C = C ( κ ) (see [29]). We are now in the position toestablish our lower bound. By simple algebraic manipulations, we have E I j j j ≈ X k ,k ′ ∈X j E β ∗ j k ( ξ j k ) β ∗ j k ′ ( ξ j k ′ ) E β ∗ j k ( ξ j k ) β ∗ j k ′ ( ξ j k ′ ) E β j k β j k ′ ( σ j σ j σ j ) × p λ j k q λ j k ′ + X k ,k ′ ∈X j (cid:0) E β j k β j k ′ − E β ∗ j k ( ξ j k ) β ∗ j k ′ ( ξ j k ′ ) (cid:1) σ j × E b β j k b β j k ′ E b β j k b β j k ′ p λ j k q λ j k ′ + X k ,k ′ ∈X j E β ∗ j k ( ξ j k ) β ∗ j k ′ ( ξ j k ′ ) (cid:0) E β j k β j k ′ − E β ∗ j k ( ξ j k ) β ∗ j k ′ ( ξ j k ′ ) (cid:1) ( σ j σ j ) × E b β j k b β j k ′ p λ j k q λ j k ′ .In view of (15), (10) and standard manipulations, the second and third compo-nents can be made arbitrarily small. To bound the first component, we recallfirst two facts involving spherical harmonics (see again ([36, 23] for details),namely the so-called Gaunt integral Z S Y l m ( x ) Y l m ( x ) Y l m ( x ) dx = r (2 l + 1)(2 l + 1)(2 l + 1)4 π (cid:18) l l l m m m (cid:19) (cid:18) l l l (cid:19) ,where for the Wigner’s 3j coefficients introduced on the right-hand side we recallthe properties X m m m (cid:18) l l l m m m (cid:19) ≡
1, (16) (cid:18) l l l (cid:19) ≥ Cl I { l + l ≥ l , l + l + l = even } . (17)Using the fact that { λ jk } are the cubature weights corresponding to the space K B j +1 , easy manipulations yield1 σ j σ j σ j B j X l l l X m m m b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) C l C l C l × X k Y l m ( ξ j k ) Y l m ( ξ j k ) Y l m ( ξ j k ) λ j k × X k ′ Y l m ( ξ j k ′ ) Y l m ( ξ j k ′ ) Y l m ( ξ j k ′ ) λ j k ′ . Lan and D. Marinucci/Needlets bispectrum = 1 σ j σ j σ j B j X l l l X m m m b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) C l C l C l × (cid:18)Z S Y l m ( x ) Y l m ( x ) Y l m ( x ) dx (cid:19) = 1 σ j σ j σ j B j X l l l b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) × C l C l C l (2 l + 1)(2 l + 1)(2 l + 1)4 π × X m m m (cid:18) l l l m m m (cid:19) (cid:18) l l l (cid:19) and using σ j ≈ B j C B j , B j i ≈ l i , i = 1 , , σ j σ j σ j B j X l l l b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) × C l C l C l (2 l + 1)(2 l + 1)(2 l + 1)4 π (cid:18) l l l (cid:19) ≈ B − j − j − j X l + l ≥ l l + l + l = even b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) > c > c := inf j j j c j j j , where c j j j := B − j − j − j X l + l ≥ l l + l + l = even b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) .It is simple to show that c > . Indeed, under (4), we have c j j j ≥ Z { u + u ≥ u } , [0 , ∩{ u ≤ u ≤ u } b ( u ) b ( u ) b ( u ) du du du ≥ Z δ − δ du Z δ u du Z − δ δ du (cid:8) b ( u ) b ( u ) b ( u ) (cid:9) ≥ c δ , where δ ≤ / , and b ( x ) ≥ / x ∈ [1 − δ , δ ] , c =inf x ∈ [1+ δ , − δ ] b ( x ) . The same argument as before could be used to establish lower bounds when j = j < j or j = j = j . To conclude our proof, we consider the case when j < j = j . We obtain . Lan and D. Marinucci/Needlets bispectrum B j − j X k k ′ E b β j k b β j k ′ (cid:16) E b β j k b β j k ′ (cid:17) × p λ j k { k , k ∈ V j k ∩ X j } p λ j k ′ { k ′ , k ′ ∈ V j k ′ ∩ X j }≥ C ( κ ) B j − j B j − j X k X k ∈V j k ∩X j (cid:16) E b β j k (cid:17) λ j k + C ′ ( κ, ρ ) B j − j B j − j X k = k ′ X k ∈V j k ∩X j ,k ′ ∈V j k ′ ∩X j (cid:16) E b β j k b β j k ′ (cid:17) ,the two summands corresponding to the cases where k , k ′ belong to the sameor to different Voronoi cells, respectively. The first part is equal to C ( κ ) B j − j X k X k ∈V j k ∩X j λ j k ≥ C ( κ ) X k λ j k = 4 πC ( κ ) ,while the second part is equal to C ′ ( κ, ρ ) B − j ( X k = k ′ ; d ( V j k , V j k ′ )=0 X k ∈V j k ∩X j ,k ′ ∈V j k ′ ∩X j + X k = k ′ ; d ( V j k , V j k ′ ) > X k ∈V j k ∩X j ,k ′ ∈V j k ′ ∩X j ) × (cid:16) E b β j k b β j k ′ (cid:17) ≤ C ′ ( κ, ρ ) 6 πB j X k X k ∈V j k ∩X j ,k ′ ∈V j k ′ ∩X j (cid:16) E b β j k b β j k ′ (cid:17) + C ′ ( κ, ρ ) B j X k X k ∈V j k ∩X j (X k ′ X k ′ ∈V j k ′ ∩X j ; d ( V j k , V j k ′ ) > (cid:16) E b β j k b β j k ′ (cid:17) ) ≤ C ′ ( κ, ρ ) 6 πB j X k B j − j + C ′ ( κ, ρ ) B j X k X k ′ C M (cid:0) B j ( B − j + d ( k ′ , V j k ( k ) )) (cid:1) M ≤ C ′ ( κ, ρ ) B − ( j − j ) + C ′ ( κ, ρ, C M ) Z ∞ B − j B j sin θ (1 + B j θ ) M dθ = C ( κ, ρ, C M ) B − ( j − j ) )By taking j − j ≥ K ≥ max (cid:8) [log B { κ/ǫ } ] , c /c (cid:9) , where c = C ( κ ) , c = C ( κ, ρ, C M ) , and ǫ ≤ c/C M , the argument is completed . The proof for case(3) is similar (actually slightly easier). . Lan and D. Marinucci/Needlets bispectrum
The following weak convergence theorem is the main result of this Section.We stress that the statement could be easily extended to a multivariate CentralLimit Theorem; however, because this extension would not entail any substantialnovelty, at the same time making the notation much more cumbersome, we preferto stick to the univariate case.
Theorem 2.
Let T ( x ) be a zero-mean, mean square continuous and isotropicGaussian random field, with angular power spectrum that satisfies Condition A.As j → ∞ , we have I j j j q EI j j j → d N (0 , .Proof. In view of the results in [32], to establish a Central Limit Theorem for amultilinear form in Gaussian random variables, it is enough to investigate theasymptotic behaviour of fourth order moments (or equivalently cumulants), seealso [14]. Our aim is then to show that, as j → ∞ , E I j j j = 3( E I j j j ) + O ( B − j / ) . For notational simplicity, we shall write ρ j ( k ′ , k ) := E b β j,k ′ b β j,k . By the diagram formula we have, for all index sets I : E (Y i ∈ I Y l =1 b β j l k il ) = X γ ∈ Γ( I, Y { ( i,l )( i ′ ,l ′ ) }∈ γ δ j l ′ j l ρ j l ( k ( i ) l , k ( i ′ ) l ′ ).Similarly to [23], we define ρ ( γ ; j , j , j ) = X k i ∈X j Y { ( i,l )( i ′ ,l ′ ) }∈ γ δ j l ′ j l ρ j l ( k ( i ) l , k ( i ′ ) l ′ ) × Y i ∈ I δ j j j ( k ( i )1 , k ( i )2 , k ( i )3 ) h j j j ( k ( i )1 , k ( i )2 , k ( i )3 )so that E I j j j = E (X k b β j k b β j k b β j k δ j j j ( k , k , k ) h j j j ( k , k , k ) ) = X k (1)3 · · · X k (4)3 E ( Y i =1 3 Y l =1 b β j l k il δ j j j ( k ( i )1 , k ( i )2 , k ( i )3 ) h j j j ( k ( i )1 , k ( i )2 , k ( i )3 ) ) = ( X γ ∈ Γ C (4 , + X γ ∈ Γ C (4 , ) ρ ( γ ; j , j , j )where Γ C is the set of all connected diagrams. To conclude our argument, weonly need to show that X γ ∈ Γ C (4 , ρ ( γ ; j , j , j ) = O ( B − j / ), as j → ∞ , (18) . Lan and D. Marinucci/Needlets bispectrum and X γ ∈ Γ C (4 , ρ ( γ ; j , j , j ) = X γ ∈ Γ P (4 , ρ ( γ ; j , j , j ) = 3( E I j j j ) . (19)(19) is an immediate consequence of the definition of E I j j j and trivial com-binatorial manipulations(see [34]). The result in (18) is proved by splitting con-nected diagrams into those with or without flat edges. Diagrams with flat edgesare dealt with in Lemma 3, while those without are dealt with in Lemma 4. Westress that, on the contrary of what is often the case when the diagram formulais applied, terms with flat edges do not vanish, due to correlation among differ-ent locations in the spherical needlets coefficients. This completes the proof ofthe main result. Lemma 3.
For a connected diagram with flat edges, γ ∈ Γ CF (4 , , we have ρ ( γ ; j , j , j ) = O ( B − j ) , as j → ∞ .Proof. We write { ( k ( a ) b , j b ) } a =1 ,..., ,b =1 , , for the elements in our diagram, a and b denoting the row and column indexes, respectively. We recall also that k = k ( k ) , k = k ( k ) , as explained earlier. For Case 1), i.e. j < j − 2, since E { β j k β j k } = 0 for every k ∈ X j , k ∈ X j , it is easy to see that ρ ( γ ; j , j , j ) ≡ . For Case 2), i.e. j + 1 < j = j , we assume (without loss ofgenerality) that a flat edge is present in the first row of the diagram, i.e. we let { ( k (1)2 , j )( k (1)3 , j ) } ∈ γ . By the same argument as in (7) we obtain immediately B j − j Y i ∈{ ,.., } X k ( i )2 ,i =1 Y { ( i,l )( i ′ ,l ′ ) }∈ γ,i,i ′ =1 δ j l ′ j l ρ j l ( k ( i ) l , k ( i ′ ) l ′ ) × X k (1)2 ρ j ( k (1)1 , k (2)1 ) q λ j k (1)1 { k (1)2 , k (1)2 ∈ V j k (1)1 ∩ X j } = 0.On the other hand, under j = j < j − , again we assume a flat edge { ( k (1)1 , j )( k (1)1 , j ) } ∈ γ ; by necessity, there should exist another flat edge in thegraph, and w.l.o.g. we take it be in the fourth row, i.e. { ( k (4)1 , j )( k (4)2 , j ) } ∈ γ .Then we have ξ j k ( i )1 = ξ j k ( i )2 , and the resulting term can be bounded by1 B j Y i ∈{ ,.., } X k ( i )3 (cid:12)(cid:12)(cid:12) ρ j ( k (1)3 , k (2)3 ) ρ j ( k (3)3 , k (4)3 ) (cid:12)(cid:12)(cid:12) ρ j ( k (2)1 , k (3)1 )= 1 B j X k (2)3 ,k (3)3 ρ j ( k (2)1 , k (3)1 ) (X k (1)3 (cid:12)(cid:12)(cid:12) ρ j ( k (1)3 , k (2)3 ) (cid:12)(cid:12)(cid:12) X k (4)3 (cid:12)(cid:12)(cid:12) ρ j ( k (3)3 , k (4)3 ) (cid:12)(cid:12)(cid:12)) ≤ B j X k (2)3 ,k (3)3 ρ j ( k (2)1 , k (3)1 ) C M × C M . Lan and D. Marinucci/Needlets bispectrum ≤ CB j X k (2)1 ,k (3)1 ρ j ( k (2)1 , k (3)1 ) " max k (2)1 n ξ j k ∈ V j ( k (2)1 ) o × " max k (3)1 n ξ j k ∈ V j ( k (3)1 ) o = CB j B j B j − j = O ( B − j ) . Finally for Case 3), the argument is analogous; more precisely, components withdiagrams in Γ CF (4 , 3) can be bounded by | ρ ( γ ; j, j, j ) | = 1 B j Y i ∈ I X k ( i ) (cid:12)(cid:12)(cid:12) ρ j ( k (1) , k (2) ) ρ j ( k (3) , k (4) ) (cid:12)(cid:12)(cid:12) ρ j ( k (2) , k (3) )= 1 B j X k (2) ,k (3) ρ j ( k (2) , k (3) ) (X k (1) (cid:12)(cid:12)(cid:12) ρ j ( k (1) , k (2) ) (cid:12)(cid:12)(cid:12) X k (4) (cid:12)(cid:12)(cid:12) ρ j ( k (3) , k (4) ) (cid:12)(cid:12)(cid:12)) ≤ B j C M X k i ,k i ρ j ( k (2) , k (3) ) = O ( B − j ) , where in the third equation we used (9). Thus the proof is completed. Lemma 4. For a connected diagram without a flat edge, γ ∈ Γ CF (4 , , we have ρ ( γ ; j , j , j ) = O ( B − j / ) , as j → ∞ .Proof. Connected diagrams without flat edges and with four nodes can be par-titioned into two classes, i.e. so-called cliques, where each vertex is connected toall three others, and terms with loops of order 2. We focus on the former class;without loss of generality, we can express ρ ( γ ; j , j , j ) by Y i =1 X k i ρ j ( k (1)1 , k (2)1 ) ρ j ( k (3)1 , k (4)1 ) ρ j ( k (1)2 , k (3)2 ) ρ j ( k (2)2 , k (4)2 ) × ρ j ( k (1)3 , k (4)3 ) ρ j ( k (2)3 , k (3)3 ) × Y i =1 δ j j j ( k ( i )1 , k ( i )2 , k ( i )3 ) h j j j ( k ( i )1 , k ( i )2 , k ( i )3 ).By means of (9) we readily obtain the bound CB j Y i =1 X k i B j d ( k (1)1 , k (2)1 )) M B j d ( k (3)1 , k (4)1 )) M × B j d ( k (1)2 , k (3)2 )) M B j d ( k (2)2 , k (4)2 )) M × B j d ( k (1)3 , k (4)3 )) M B j d ( k (2)3 , k (3)3 )) M . (20) . Lan and D. Marinucci/Needlets bispectrum Consider first Case 2), with j + 1 < j = j . Using inequality (11), (20) can bereplaced by B j − j X k (1)2 ,..,k ( I )2 ∈X j B j d ( k (1)2 , k (3)2 )) M B j d ( k (2)2 , k (4)2 )) M × B j d ( k (1)2 , k (4)2 )) M B j d ( k (2)2 , k (3)2 )) M × Y i =1 q λ j k ( i )2 { k ( i ) , k ( i ) ∈ V j k ( i )1 ∩ X j }≤ B j CB j X k (3)2 ,k (4)2 ( X k (1)2 B j d ( k (1)2 , k (3)2 )) M B j d ( k (1)2 , k (4)2 )) M ) × ( X k (2)2 B j d ( k (2)2 , k (4)2 )) M B j d ( k (2)2 , k (3)2 )) M ) ≤ CB j X k (3)2 ,k (4)2 (cid:26) C M (1 + B j d ( k (3)2 , k (4)2 )) M (cid:27) ≤ CB j C M C M B j = O ( B − j ) = o ( B − j ).Likewise, for j = j < j − , we have the bound CB j Y i =1 X k ( i )3 B j d ( k (1)1 , k (2)1 )) M B j d ( k (3)1 , k (4)1 )) M × B j d ( k (1)1 , k (3)1 )) M B j d ( k (2)1 , k (4)1 )) M B j d ( k (1)3 , k (4)3 )) M × B j d ( k (2)3 , k (3)3 )) M ≤ CB j Y i =1 X k ( i )1 B j d ( k (1)1 , k (2)1 )) M B j d ( k (3)1 , k (4)1 )) M × B j d ( k (1)1 , k (3)1 )) M B j d ( k (2)1 , k (4)1 )) M × X k ∈V j ( k (1)1 ) ∩X j B j d ( k (1)3 , k (4)3 )) M × X k ∈V j ( k (2)1 ) ∩X j B j d ( k (2)3 , k (3)3 )) M . Lan and D. Marinucci/Needlets bispectrum ≤ CB j − j B j X k (2)1 ,k (3)1 (X k (1)1 B j d ( k (1)1 , k (2)1 )) M B j d ( k (1)1 , k (3)1 )) M (cid:27) × (X k (4)1 B j d ( k (3)1 , k (4)1 )) M B j d ( k (2)1 , k (4)1 )) M ) ≤ CB j − j B j X k (2)1 ,k (3)1 C M (1 + B j d ( k (2)1 , k (3)1 )) M ≤ CB j − j = O ( B − j ).This concludes the proof for Case 2); the proof for Case 3) could be implementedalong identical lines.The analysis for case 1) j + 1 < j < j − , i.e. the case where all threefrequencies differ, requires considerably more care. As before, let V j u ( k ( i ) u , r )be the Voronoi cells associated to k ( i ) u , and we recall it satisfies B ( x i , r/ ⊂V j u ( x i ) ⊂ B ( x i , r ), k ( i ) u ∈ X j u . Our idea is to partition (20) into four elements,as follows:(20) ≤ CB j Y i =1 X k i { I ( d ( k (1)1 , k (4)1 > r ) , d ( k (2)1 , k (3)1 > r ))+ I ( d ( k (1)1 , k (4)1 > r ) , d ( k (2)1 , k (3)1 ≤ r ))+ I ( d ( k (1)1 , k (4)1 ≤ r ) , d ( k (2)1 , k (3)1 > r ))+ I ( d ( k (1)1 , k (4)1 ≤ r ) , d ( k (2)1 , k (3)1 ≤ r )) }× B j d ( k (1)1 , k (2)1 )) M B j d ( k (3)1 , k (4)1 )) M B j d ( k (1)2 , k (3)2 )) M × B j d ( k (2)2 , k (4)2 )) M B j d ( k (1)3 , k (4)3 )) M B j d ( k (2)3 , k (3)3 )) M The first three summands are easy to bound, indeed it is enough to notice that CB j Y i =1 X k i n I (cid:16) d (cid:16) k (1)1 , k (4)1 > r (cid:17) , d (cid:16) k (2)1 , k (3)1 > r (cid:17) (cid:17)o × B j d ( k (1)1 , k (2)1 )) M B j d ( k (3)1 , k (4)1 )) M B j d ( k (1)2 , k (3)2 )) M × B j d ( k (2)2 , k (4)2 )) M B j d ( k (1)3 , k (4)3 )) M B j d ( k (2)3 , k (3)3 )) M ≤ CB j Y i =1 X k i n I (cid:16) d (cid:16) k (1)1 , k (4)1 > r (cid:17) (cid:17)o B j d ( k (1)3 , k (4)3 )) M × B j d ( k (2)3 , k (3)3 )) M . Lan and D. Marinucci/Needlets bispectrum ≤ CB j B j X k (1)3 ,k (4)3 I (cid:16) d ( k (1)1 , k (4)1 > r ) (cid:17) B j d ( k (1)3 , k (4)3 )) M ≤ CB j B j − j B j = O ( B j − j ) = O ( B − j )because X k ′ ∈Z j I ( d ( k, k ′ ) > r ) 1(1 + B j d ( k, k ′ )) M ≤ CB j Z πr sin θ (1 + B j θ ) M dθ ≤ CB j Z πr θdθB Mj θ M ≤ CB ( M − j − j ) ,by taking r = B − j / , compare Lemma 10 in [6]. The argument for the secondand third term is analogous. Concerning the last summand, we recall that Card { k ′ ∈ V j / ( x i ) ∩ X j } ≈ B j − j / , (21)for every k ∈ X j / . Now denoteΩ( k ; j ) := { ( k , k , k , k ) : k , k , k , k ∈ X j , k ∈ V j / ( k ) , d ( k , k ) ≤ r, d ( k , k ) ≤ r } ,where r = κB − j . Heuristically, the idea is to split Ω( k ) into regions where( k , k , k , k ) are each “close” to all three others, and regions where they areclose two by two but the two pairs need not belong to the same neighbourhood.More precisely, we take Ω( k ; j ) ⊆ ∆ ∪ ∆ , where∆ = { ( k , k , k , k ) : k , . . . , k ∈ B r ( k ) , k ∈ V j / ( k ) } and ∆ = { ( k , k , k , k ) : k ∈ V j / ( k ) , k ∈ S /B r ( k ) } . We can hence defineΩ ( k ; j ) := Ω( k ; j ) ∩ ∆ , Ω ( k ; j ) := Ω( k ; j ) ∩ ∆ . In the region Ω ( k ; j ) , the idea is to keep k fixed and then proceed by evaluatingthe cardinality of B r ( k ); in view of (21), this leads to(20) ≤ CB j X k ∈X j / X ( k ,k ,k ,k ) ∈ Ω ( k ; j ) , B j d ( k , k )) M B j d ( k , k )) M = O (cid:18) B j − j / j B j (cid:19) = O ( B − j ) , . Lan and D. Marinucci/Needlets bispectrum by (9) and Lemma 4.8 in [31]. On the other hand, in the region Ω ( k ), weexploit the fact that d ( k , k ) , d ( k , k ) ≥ r = 3 κB − j / , so that we obtain theupper bound (for some C > ≤ CB j X k ∈X j / X ( k (1)3 ,...,k (4)3 ) ∈ Ω2( k ; j B j d ( k (1)2 , k (2)2 )) M × B j d ( k (3)2 , k (4)2 )) M B j d ( k (1)3 , k (4)3 )) M B j d ( k (2)3 , k (3)3 )) M ≤ CB j X k ∈X j X ( k ,k ,k ,k ) ∈ Ω ( k ; j ) C ( B j − j / )) M B j d ( k , k )) M × B j d ( k , k )) M = O (cid:18) B j − M ( j − j / B j (cid:19) = O ( B − Mj ) ).The proof for the remaining terms is very similar and hence omitted for brevity’ssake. As the final result of this Section, we wish to focus on the case where the varianceof the needlets coefficients is unknown and estimated from the data. A naturalestimator for σ j is provided by e σ j = 1 N j X ξ jk ∈X j | β jk | where as before N j = card {X j } ≈ B j . We define our studentized statistics as e β jk := β jk / e σ j and we then consider e I j j j = X k e β j k e β j k e β j k δ j j j ( k , k , k ) h j j j ( k , k , k ).Our next result shows that this studentization procedure has no effect on asymp-totic behaviour. Theorem 5. As j → ∞ , we have (cid:8) σ j σ j σ j (cid:9) − e σ j e σ j e σ j −→ in probabil-ity, and hence e I j j j → d N (0 , .Proof. By a standard application of Slutzki’s theorem, the weak convergenceresult follows immediately from the consistency of the variance estimator. Weprovide a proof for the three cases separately. . Lan and D. Marinucci/Needlets bispectrum For case 1), i.e. j < j < j we have immediately E [ (cid:8) σ j σ j σ j (cid:9) − e σ j e σ j e σ j ] =1. Moreover V ar [ e σ j e σ j e σ j ] = 1 N j N j N j Y i =1 X ξ jiki ,ξ jik ′ i ∈X ji E [ | β j i k i | | β j i k ′ i | ] − σ j σ j σ j ,where X ξ j k ,ξ j k ′ ∈X j E [ | β j k | | β j k ′ | ] = X ξ j k ∈X j E | β j k | ! + 2 X ξ j k ,ξ j k ′ ∈X j | E [ β j k , β j k ′ ] | = N j σ j + O (cid:0) N j σ j (cid:1) .Hence V ar [ e σ j e σ j e σ j ] = 1 N j N j N j ( N j + O ( N j ))( N j + O ( N j )) × ( N j + O ( N j )) σ j σ j σ j − σ j σ j σ j = O σ j σ j σ j N j N j N j ! = O (cid:18) N j N j N j (cid:19) .For case 2), we focus on the case where j = j < j ; the remaining part of theproof is nearly identical. First note that E ( e σ j e σ j σ j σ j ) = P ξ j k ∈X j E | β j k | N j σ j σ j × " πN j X ξ j k ∈X j E | β j k | ! + 2(4 π ) N j X ξ j k ,ξ j k ′ ∈X j | E [ β j k , β j k ′ ] | = 1 σ j σ j ( σ j + O σ j N j ! ) σ j = 1 + O (cid:18) N j (cid:19) .Likewise V ar [ e σ j e σ j ] = E (cid:18) N j X ξ j k ∈X j | β j k | (cid:19) E (cid:18) N j X ξ j k ∈X j | β j k | (cid:19) − σ j + O σ j N j !! σ j . Lan and D. Marinucci/Needlets bispectrum = 1 N j N j E " Y i =1 X ξ j kI ∈X j | β j k | | β j k | | β j k | | β j k | × E X ξ j k ∈X j | β j k | ! − (cid:18) σ j + O σ j N j !(cid:19) σ j = " X ξ j k ∈X j E | β j k | ! + 12 X ξ j k ∈X j E | β j k | ! × X ξ j k ,ξ j k ′ ∈X j | E [ β j k , β j k ′ ] | + 8 X ξ j k ∈X j E | β j k | ! × Y i =1 X ξ j ki ∈X j E [ β j k , β j k ] E [ β j k , β j k ] E [ β j k , β j k ] ! × σ j (1 + O (cid:16) N j (cid:17) ) N j − (cid:18) σ j + O σ j N j !(cid:19) σ j .By (9) and [31] (Lemma 4.8), we have Y i =1 X ξ j kI ∈X j E [ β j k , β j k ] E [ β j k , β j k ] E [ β j k , β j k ≤ σ j Y i =1 X ξ j kI ∈X j B j d ( k , k )) B j d ( k , k )) B j d ( k , k )) ≤ σ j X ξ j k ,ξ j k ∈X j C M (1 + B j d ( k , k )) ≤ C M N j σ j .Hence the variance is bounded by (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) N j σ j + 8 σ j O (cid:0) N j (cid:1) + 12 σ j O (cid:0) N j (cid:1)(cid:3) × σ j (cid:0) O (cid:0) N j (cid:1)(cid:1) N j − (cid:18) σ j + O (cid:18) σ j N j (cid:19)(cid:19) σ j (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) σ j σ j (cid:18) N j + 1 N j (cid:19)(cid:19) = O (cid:18) N j (cid:19) .The proof for case 3) is an easy consequence of results by [6, 7]. 4. Convergence to multiparameter Gaussian processes Our aim in this Section is to extend the previous results to functional conver-gence theorems . The motivation for such an extension can be easily explained. . Lan and D. Marinucci/Needlets bispectrum Indeed, from the applications points of view, practitioners are typically inter-ested not only at the possible existence of non-Gaussianity and/or other fea-tures, but also to their location in frequency space. If we focus for instanceon cosmological applications, which are the main driving rationale behind ourwork, it is important to recall that the existence of possible non-Gaussianitiestakes a very different meaning according to the scales where they are located,so that a suitable statistical procedure should provide information not only ontheir existence, but also on their position in the frequency domain. As an ex-ample, a huge debate has arisen in the Cosmological literature on the possibleexistence of a non-Gaussian “Cold Spot” in CMB data, much of the related lit-erature concerning the determination of the angular scale of such features, seefor instance [12, 13]. Concerning this feature, it may even be of interest to testfor Gaussianity only on a subspace of the sphere (this is indeed what happensin practice, because of missing observations). The modification of (13) underthese circumstances is straightforward: we would simply restrict our sum to asubset of the cubature points. Our following discussion would be asymptoticallyunaltered.In [11, 23], it was proposed to build alternative forms of partial sum processfrom the bispectrum B l l l , and to use them as a probe of non-Gaussianity atvarious scales. All different proposals were univariate, in the following sense. As-sume the resolution of the experiment is such that frequencies up to l = 1 , . . . , L are observed; the partial sums were then run only on a subset of configurationswith cardinality L , whereas the number of multipole combinations ( l , l , l )which would be available for statistical analysis is in the order of L . One ofthe reasons for this restriction had to do with computational complexity: theevaluation of even a single bispectrum statistic is extremely time consuming,so that the exploration of all possible configurations is likely to be unfeasibleeven on the greatest supercomputing facilities. On the contrary, needlets areextremely convenient from a computational point of view, and there is no ob-stacle in considering larger frequency configurations, provided of course that thetriangle conditions are satisfied.We shall hence focus on two possible partial sums processes, which correspondbroadly to cases 1 and 2 of the previous section; more precisely J L ( r , r ) = 1 L [ Lr ] X j =1 [ Lr ] X m =0 b I j ,j + K + m,j + K + m , (22) J L ( r ) = 1 √ L qP N m =0 ( N ( m ) + 1) [ Lr ] X j =1 N X m =0 N ( m ) X m =0 b I j ,j + K + m , j +2 K + m + m ,(23)where b I j ,j ,j := I j ,j ,j q EI j ,j ,j , . Lan and D. Marinucci/Needlets bispectrum K ≥ N = max (cid:8) m : 1 + B K + m ≥ B K + m (cid:9) N ( x ) := max (cid:8) x : log B (cid:0) B K + x (cid:1) − K − m ≥ (cid:9) . Theorem 6. a) As L → ∞ J L ( r ,r ) ⇒ W ( r , r ) , in D [0 , , (24) where W ( . ) is two-dimensional Brownian sheet, i.e. the zero mean Gaussianprocess with covariance function EW ( r , r ) W ( s , s ) = ( r ∧ s )( r ∧ s ) . b) As L → ∞ J L ( r ) ⇒ W ( r ) , in D [0 , 1] (25) where W ( . ) is standard Brownian Motion.Here, ⇒ denotes weak convergence in the sense of [9], D [0 , p , p ∈ N is theusual multidimensional Skorohod space.Proof. We start from (24); as usual, we need to establish convergence of thefinite-dimensional distributions and tightness. Obviously, EJ L ( r , r ) = 0 , and because E [ ˆ I j ,j ,j ˆ I j ′ ,j ′ ,j ′ ] = δ j ′ j δ j ′ j , EJ L ( r , r ) J L ( s , s )= 1 L Lr ] X j =1 [ Lr ] X m =0 [ Ls ] X j ′ =1 [ Ls ] X m ′ =0 E ˆ I j ,j + K + m,j + K + m ˆ I j ,j + K + m ′ ,j + K + m ′ = 1 L Lr ] ∧ [ Ls ] X j =1 [ Lr ] ∧ [ Ls ] X m =0 E b I j ,j + K + m,j + K + m = [ Lr ] ∧ [ Ls ] X j =1 [ Lr ] ∧ [ Ls ] L → [ r ∧ s ] [ r ∧ s ] .To establish Gaussianity, we can again rely on the results by [32] and proceedwith the bounds for the fourth-order cumulants. As the computations are verymuch the same as in the previous Section, we omit the details for brevity’s sake.To consider tightness, we use the classical criteria given for instance in [35].Define first the two-dimensional increments J L (( s , r ] × ( s , r ]) := J L ( r , r ) − J L ( r , s ) − J L ( s , r ) + J L ( s , s ). . Lan and D. Marinucci/Needlets bispectrum It is again a standard computation to show that, as L → ∞ ,EJ L (( s , r ] × ( s , r ]) = 1 L ( [ Lr ] X j =[ Ls ]+1 [ Lr ] X m =[ Ls ]+1 E ˆ I j ,j + K + m,j + K + m ) = 1 L { [ Lr ] − [ Ls ] }{ [ Lr ] − [ Ls ] }≤ { r − s }{ r − s } . We can then establish tightness by showing that1) s i ≤ t i ≤ r i , i = 1 , E [ J L (( s , t ] × ( s , t ]) J L (( t , r ] × ( t , r ])]= 1 L E [ Lt ] X j ,j ′ =[ Ls ] [ Lt ] X j = j + K + m,m =[ Ls ]+1 [ Lt ] X j ′ = j ′ + K + m ′ ,m ′ =[ Ls ]+1 ˆ I j j j ˆ I j ′ j ′ j ′ ! × [ Lr ] X j ,j ′ =[ Lt ] [ Lr ] X j = j + K + m,m =[ Lt ]+1 [ Lr ] X j ′ = j ′ + K + m ′ ,m ′ =[ Lt ]+1 ˆ I j j j ˆ I j ′ j ′ j ′ ! = 1 L [ Lt ] X j =[ Ls ] [ Lt ] X m =[ Ls ] E ˆ I j ,j + K + m,j + K + m ! × [ Lr ] X j ′ =[ Lt ] [ Lr ] X m ′ =[ Lt ] E ˆ I j ′ ,j ′ + K + m ′ ,j ′ + K + m ′ ! ≤ r − t )( r − t )( t − s )( t − s ) ≤ r − s ) ( r − s ) . s ≤ r , s ≤ t ≤ r ,E [ J L (( s , r ] × ( s , t ]) J L (( s , r ] × ( t , r ])] = 1 L E [ Lt ] X j ,j ′ =[ Ls ] [ Lt ] X j = j + K + m,m =[ Ls ]+1 [ Lt ] X j ′ = j ′ + K + m ′ ,m ′ =[ Ls ]+1 ˆ I j j j ˆ I j ′ j ′ j ′ ! × [ Lr ] X j ,j ′ =[ Ls ] [ Lr ] X j = j + K + m,m =[ Lt ]+1 [ Lr ] X j ′ = j ′ + K + m ′ ,m ′ =[ Lt ]+1 ˆ I j j j ˆ I j ′ j ′ j ′ ! = 1 L [ Lr ] X j =[ Ls ] [ Lt ] X m =[ Ls ] E ˆ I j ,j + K + m,j + K + m ! × [ Lr ] X j ′ =[ Ls ] [ Lr ] X m ′ =[ Lt ] E ˆ I j ′ ,j ′ + K + m ′ ,j ′ + K + m ′ ! . Lan and D. Marinucci/Needlets bispectrum + 1 L Lr ] X j ( i )1 =[ Ls ]+1 ,i =1 ,.., Lt ] X m ( i ) =[ Ls ]; i =1 , Lr ] X m ( i ) =[ Lt ]; i =3 , × X γ ∈ Γ C ρ γ ; Y i =1 b I j ( i )1 ,j ( i )1 + K + m ( i ) ,j ( i )1 + K + m ( i ) ! For the first part it is easy to see that it is bounded by 16( r − s ) ( r − t )( t − s ); for the second part, for j ( i )1 , j ( i )2 in each of their domain, we have, X γ ∈ Γ C ρ (cid:18) γ ; Y i =1 b I j ( i )1 j ( i )2 j ( i )2 (cid:19) = X γ ∈ Γ C ρ ( γ ; b I j j (1)2 j (1)2 b I j j (3)2 j (3)2 ) Y i =1 δ j ( i )1 j δ j (2)2 j (4)2 j (1)2 j (3)2 + X γ ∈ Γ C ρ ( γ ; b I j j ′ j ′ b I j ′ j (3)2 j (3)2 ) Y i =1 δ j ( i )1 j ( i )2 δ j (2)2 j (4)2 j (1)2 j (3)2 ,where Γ C denotes the graphs with cliques (all nodes connected with all others),where Γ C refers to graphs with loops of order two; these two disjoint classescover all possible connected graphs with four nodes. We have1 L Lr ] X j ( i )1 =[ Ls ]+1 ,i =1 ,.., Lt ] X m ( i ) =[ Ls ]; i =1 , Lr ] X m ( i ) =[ Lt ]; i =3 , X γ ∈{ fig } ρ (cid:18) γ ; Y i =1 b I j ( i )1 ,j ( i )1 + K + m ( i ) ,j ( i )1 + K + m ( i ) (cid:19) ≤ CL Lr ] X j , j ′ =[ Ls ]+1 [ Lt ] X m ( i ) =[ Ls ]; i =1 , Lr ] X m ( i ) =[ Lt ]; i =3 , ≤ C ′ ( r − s ) ( r − t )( t − s )from which we obtain E [ J L (( s , r ] × ( s , t ]) J L (( s , r ] × ( t , r ])] ≤ C ( r − s ) ( r − t )( t − s )Similarly, we can get the same result for3) s ≤ t ≤ r , s ≤ r , that is E [ J L (( s , t ] × ( s , r ]) J L (( t , r ] × ( s , r ])] ≤ C ( r − s ) ( r − t )( t − s )This concludes the proof of (24).For (25), we start again from the convergence of the finite-dimensional distri-butions; for notational simplicity, we stick to the univariate case. It is obvious . Lan and D. Marinucci/Needlets bispectrum that EJ L ( r ) = 0; on the other hand, EJ L ( r ) J L ( s )= [ Lr ] X j =1 [ Ls ] X j ′ =1 N X m ,m ′ =0 N ( m ) X m =0 N ( m ′ ) X m ′ =0 × E b I j ,j + K + m ,j +2 K + m + m b I j ′ ,j ′ + K + m ′ ,j ′ +2 K + m ′ + m ′ L P N m =0 ( N ( m ) + 1)= 1 L P N m =0 ( N ( m ) + 1) [ Lr ] ∧ [ Ls ] X j =1 N X m =0 N ( m ) X m =0 E b I j ,j + K + m ,j +2 K + m + m = 1 L ([ Lr ] ∧ [ Ls ] − → [ r ∧ s ] , if min { r, s } > . For Gaussianity, we analyze once again fourth-order cumulants, i.e. the con-nected components in the expansion of the fourth moment. As before, we needonly focus on connected diagrams with four nodes, which can be partitioned intotwo classes: the cliques, where all nodes are connected with all three others, anddiagrams with a loop of order 2. As before, these terms can be bounded by ρ (cid:18) γ ; Y i =1 b I j ( i )1 j ( i )2 j ( i )3 (cid:19) = O ( B − max { j ( i )1 } / ),because for instance1 B j (1)3 + j (2)3 +2 j (3)3 X k (1)3 ,k (2)3 ,k (3)3 ,k (4)3 B j (1)1 d ( k (1)1 , k (2)1 )) M B j (1)2 d ( k (1)2 , k (4)1 )) M × B j (2)2 d ( k (2)2 , k (3)1 )) M B j (1)3 d ( k (1)3 , k (3)2 )) M B j (2)3 d ( k (2)3 , k (4)2 )) M × B j (3)3 d ( k (3)3 , k (4)3 )) M ≤ P k (3)3 ,k (4)3 B j (3)3 d ( k (3)3 ,k (4)3 )) M B j (1)3 + j (2)3 +2 j X k (1)3 B j (1)3 d ( k (1)3 , k (3)2 )) M × X k (2)3 B j (2)3 d ( k (2)3 , k (4)2 )) M ≤ B j (1)3 + j (2)3 +2 j X k (3)3 ,k (4)3 C M (1 + B j (3)3 d ( k (3)3 , k (4)3 )) M ≈ C M B j B j (1)3 + j (2)3 +2 j = O ( B − ( j (1)3 + j (2)3 ) ). . Lan and D. Marinucci/Needlets bispectrum To sum up Y i =1 [ Lr ] X j ( i )1 =1 N X m ( i )1 =0 N (cid:0) m ( i )1 (cid:1)X m ( i )2 =0 X γ ∈ Γ C ρ (cid:18) γ ; Y i =1 b I j ( i )1 ,j ( i )1 + K + m ( i )1 ,j ( i )1 +2 K + m ( i )1 + m ( i )2 (cid:19) ≤ C [ Lr ] N X m =0 ( N ( m ) + 1) ! Lr ] X j =1 B − j / = O ( L ).It follows easily that EJ L ( r ) = [ Lr ] X j =1 N X m =0 N ( m ) X m =0 (cid:0) E ˆ I j ,j + K + m ,j +2 K + m + m (cid:1) L (cid:16)P N m =0 ( N ( m ) + 1) (cid:17) + Y i =1 [ Lr ] X j ( i )1 =1 N X m ( i )1 =0 N (cid:0) m ( i )1 (cid:1)X m ( i )2 =0 × P γ ∈ Γ C ρ ( γ ; Q i =1 b I j ( i )1 ,j ( i )1 + K + m ( i )1 ,j ( i )1 +2 K + m ( i )1 + m ( i )2 ) L (cid:16)P N m =0 ( N ( m ) + 1) (cid:17) = 3 | EJ L ( r ) | + O (cid:18) L (cid:19) ,which is enough to conclude the proof for the finite-dimensional distributions,in view of the standard argument from [32] that we used before.To conclude the proof, we need only to consider tightness in D ([0 , . Notethat E [ ˆ I j ,j ,j ˆ I j ′ ,j ′ ,j ′ ] = δ j ′ j δ j ′ j δ j ′ j , the variance of J L in [ s, r ] is provided by E | J L ( r ) − J L ( s ) | = [ Lr ] X j =[ Ls ]+1 N X m =0 N ( m ) X m =0 E ˆ I j ,j + K + m ,j +2 K + m + m L P N m =0 ( N ( m ) + 1) ≤ r − s )Now we establish our tightness criterion. For any 0 ≤ s ≤ t ≤ r ≤ , E | J L ( r ) − J L ( t ) | | J L ( t ) − J L ( s ) | = [ Lr ] X j =[ Lt ] N X m =0 N ( m ) X m =0 E b I j ,j + K + m , j +2 K + m + m L P N m =0 ( N ( m ) + 1) × [ Lt ] X j ′ =[ Ls ] N X m ′ =0 N ( m ′ ) X m ′ =0 E b I j ′ ,j ′ + K + m ′ , j ′ +2 K + m ′ + m ′ L P N m =0 ( N ( m ) + 1)+ [ Lr ] X j ( i )1 =[ Lt ] ,i =1 , Lt ] X j ( i )1 =[ Ls ] ,i =3 , Y l =1 N X m ( l )1 =0 N ( m ( l )1 ) X m ( l )2 =0 . Lan and D. Marinucci/Needlets bispectrum × X γ ∈ Γ C ρ ( γ ; Q l =1 b I j ( i )1 ,j ( l )1 + K + m ( l )1 ,j ( l )1 +2 K + m ( l )1 + m ( l )2 ) (cid:16) L P N m =0 ( N ( m ) + 1) (cid:17) ≤ L ([ Lr ] − [ Lt ])([ Lt ] − [ Ls ]) ≤ r − s ) .Thus we finished the proof of tightness. 5. Behaviour under non-Gaussianity In this final Section, we shall provide some quick and informal discussion onthe behaviour of our statistics under non-Gaussianity; see [33, 26] for otherapplications of the needlets to cosmological data analysis. There exist of coursea huge variety of non-Gaussian models for spherical random fields, and we shalldelay a much more detailed treatment to future work. Our purpose here isdifferent, i.e. we want to provide some heuristic discussion on the expectedbehaviour of our procedures for physically motivated non-Gaussian models. Thiswill provide some guidance to practitioners for applications to CMB data, whichare currently under way in a separate work.We start from the expected value of the needlets bispectrum, which is pro-vided by E I j j j ≈ √ B j X k k k E b β j k b β j k b β j k δ j j j ( k , k , k ) ≈ σ j σ j σ j √ B j X k k k X l ,l ,l m m m b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) × b (cid:18) l B j (cid:19) E ( a l m a l m a l m ) × Y l m ( ξ j k ) Y l m ( ξ j k ) Y l m ( ξ j k ) δ j j j ( k , k , k )= 1 σ j σ j σ j B j B j +1 X l ,l ,l = B j − X m m m b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b l l l × (cid:18) l l l m m m (cid:19) (cid:18) l l l (cid:19) r (2 l + 1)(2 l + 1)(2 l + 1)4 π × ( B j X k k k Y l m ( ξ j k ) Y l m ( ξ j k ) Y l m ( ξ j k ) δ j j j ( k , k , k ) ) .Here, we recall that b l l l is the so-called reduced bispectrum (see for instance[20, 23]), which collects the non-Gaussian component in the third order moment . Lan and D. Marinucci/Needlets bispectrum E ( a l m a l m a l m ) , i.e. by definition E ( a l m a l m a l m )= (cid:18) l l l m m m (cid:19) (cid:18) l l l (cid:19) r (2 l + 1)(2 l + 1)(2 l + 1)4 π b l l l .In the cosmological literature, a very popular model for b l l l is provided bythe so-called Sachs-Wolfe bispectrum ([20], equation (21)), which yields b l l l = − f NL { C l C l + C l C l + C l C l } ,where f NL is a physical constant (see for instance [8, 37]). The Wigner coeffi-cients on the right hand side ensure that the expected value E ( a l m a l m a l m )is rotationally invariant under a change of coordinate, an obvious consequenceof the isotropy of the random field. For our purposes below, it is sufficient torecall that (cid:18) l l l (cid:19) ≈ ( − − l/ l , (cid:18) l l l + l (cid:19) ≈ ( − − l + l √ l . (26)Now exploiting again the cubature formula (3) we obtain as before ( B j X k k k Y l m ( ξ j k ) Y l m ( ξ j k ) Y l m ( ξ j k ) δ j j j ( k , k , k ) ) ≈ Z S Y l m ( ξ ) Y l m ( ξ ) Y l m ( ξ ) dξ = (cid:18) l l l m m m (cid:19) (cid:18) l l l (cid:19) r (2 l + 1)(2 l + 1)(2 l + 1)4 π ,whence E I j j j = 1 σ j σ j σ j B j B j +1 X l ,l ,l = B j − b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b l l l (cid:18) l l l (cid:19) × (2 l + 1)(2 l + 1)(2 l + 1)4 π X m m m (cid:18) l l l m m m (cid:19) = B j σ j σ j σ j B j +1 X l ,l ,l = B j − b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b l l l (cid:18) l l l (cid:19) × (2 l + 1)(2 l + 1)(2 l + 1)4 π ,in view of the orthonormality properties of the Wigner’s 3 j coefficients. To keepthe analogy with the cosmological literature, we shall focus on “equilateral” and . Lan and D. Marinucci/Needlets bispectrum “squeezed” configurations, see [3, 23]. In the equilateral case j = j = j = j we have E I jjj = B j σ j σ j σ j B j +1 X l ,l ,l = B j − b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b l l l × (cid:18) l l l (cid:19) (2 l + 1)(2 l + 1)(2 l + 1)4 π .Now recall l ≈ B j , σ j ≈ C / B j B j , b l l l ≈ f NL l − α l − α so that, using also (17) B j B j +1 X l ,l ,l = B j − b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b l l l σ j (cid:18) l l l (cid:19) × (2 l + 1)(2 l + 1)(2 l + 1)4 π ≃ B j B j +1 X l ,l ,l = B j − b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) f NL l − α l − α C / B j B j (cid:18) l l l (cid:19) B j ≃ B j B j +1 X l ,l ,l = B j − b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) f NL B − jα B − jα/ B j B j ≃ f NL B j B − jα/ . This suggests the expected value of the needlets bispectrum can either divergeor converge to zero, according to the asymptotic behaviour of the angular powerspectrum; in particular, it does diverge for all α < . On the other hand, for j << j = j by an analogous argument we obtain E I j j j ≃ f NL B j B − j ( α/ − B − j ( α − × B j +1 X l ,l ,l = B j − b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) l − α l − α (cid:18) l l l (cid:19) B j B j ≃ f NL B j B − j ( α/ − B − j ( α − × B j +1 X l ,l ,l = B j − b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) b (cid:18) l B j (cid:19) B − j α B − j α B j ≃ f NL B − j α/ B j . As for the usual bispectrum, the previous computations suggest that the poweris maximized by “squeezing” frequencies, i.e. maximizing the differences betweenthe “side lengths” j and j . This is the same sort of qualitative result whichwas found for the bispectrum in [23] and successfully applied to CMB datain [11]. 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