On The Dependence Structure of Wavelet Coefficients for Spherical Random Fields

On The Dep endence Structure of W a v elet Co efficien ts for Spherical Random Fields ∗ Xiaohong Lan Institute of Mathematics, Chinese Academ y of Sciences and Departmen t of Mathematics, Univ ersit y of Rome T or V ergata Email: lan@mat.uniroma2.it Domenico Marin ucci Departmen t of Mathem atics, Univ ersit y of Rome T or V erg ata Email: marin ucc@mat.uniroma2.it Octob er 27, 2018 Abstract W e consider the correlation structure of the rand om co efficients for a class of w av elet systems on th e sphere (lab elled Mexican needlets) which w as recentl y in tro duced in the literature by [15]. W e pro vid e necessary and sufficient conditions for these co efficients to b e as y m p totic uncorrelated in the real and in the frequency d omain. Here, the asymptotic theory is developed in the high frequency sense. Statistical applications are also discussed, in particular with reference to th e analysis of Cosmological data. • Keywords and Phrases: Spherical Random Fields, W a velets, Mexican Needlets, H igh F requ ency Asymptotics, Cosmic Microw a ve Bac kground Radiation • AMS 2000 Classification: Primary 60G60 ; Secondary 62M40, 42C40, 42C10 1 In tro duction There is currently a rapidly gro wing liter ature on the cons truction of w av elets systems on the sphere, see for instance [14], [2], [7], [19], [30], [35] and the references therein. Some of these attempts have bee n explicitly motiv ated by extremely in teresting applications, for instance in the framework of Astronomy ∗ W e are v ery grateful to a referee and an Associ ate Editor for their very constructiv e remarks. The research of Xiaohong Lan was supp orted by the E qual Opp ortunity Committee of the Unive rs it y of Rome T or V ergat a. 1 and/or Cosmo logy; concerning the la tter, sp ecial emphasis ha s b een devoted to wa velet tec hniques for the statistical study of the Cosmic Microwa ve Bac kgro und (CMB) ra diation ([25]). CMB can b e viewed as providing observ ations o n the Univ er se in the immediate a djacency of the Big Bang, a nd as such it has b een the ob ject of immense theoretica l a nd applied in terest ov er the las t deca de [9]. Among spherica l wa velets, particular attent ion has b een devoted to so - called needlets, whic h were intro duce d into the F unctional Analy s is literature by [26, 2 7]; their statistica l prop erties were first considered by [3 ], [4], [13] and [21]. In par ticula r, it has been shown in [3] that random needlet co efficients enjoy a capital uncorrela tion proper t y: namely , for any fixed angular distance, random needlets co e fficien ts a re asymptotically uncorrelated as the frequency parameter g rows larg er and larger. The meaning of this uncorrelatio n proper t y m ust be carefully under s to o d, given the s pecific setting of statistical inference in Cosmolog y . Indeed, CMB can be viewed as a single r e alization of an isotropic random field o n a sphere of a finite radius ([9]). The asymptotic theory is then ent er tained in the high frequency sense, i.e. it is consider ed that obser v ations at hig her and hig her frequencies (sma ller and smaller s cales) b ecome av a ilable with the gr owing sophistica tion of CMB satellite exp eriments. Of cour se, unco r- relation ent a ils indep endence in the Gaus s ian case: as a consequence, from the ab ov e-mentioned prop erty it follows that an increasing array of asymptotically i.i.d. co efficients ca n b e derived out o f a single realization of a spher ical rando m field, making thus p os s ible the introduction of a v ariety o f statistica l pr o cedures for testing non-Gaussianity , estimating the angular p ow er s pectr um, testing for asymmetries, implementing b o otstr ap techniques, testing for cross-co rrelation among CMB and Large Scale Structure data, a nd many others, see fo r insta nce [3], [4], [17], [21], [13], [2 9], [23], [8], [5], [20], [31]. W e note that the r elev ance of high frequency a symptotics is not s p ecific to Cosmology (cf. e.g. financial data). Given such a widespread a rray of techniques which ar e made feasible b y means of the uncorrelation proper t y , it is natural to in vestigate to what extent this pro p er ty should b e considered uniq ue for the construction in [26, 27], or else whether it is actually shared b y other prop osals. In particula r , we shall fo cus here on the approach which ha s bee n very rece ntly advocated by [15], see also [14] for a related setting. This approach (which we shall discuss in Section 2) can b e la belled Mexican needlets, for reaso ns to be made clear later. Its analysis is ma de pa rticularly in teresting b y the fact that, as we shall discuss below, the Mexican needlets can b e consider ed asymptotically equiv alent to the Spherical Mexican Hat W av elet (SMHW), which is curr en tly the most p opular wa velet pro cedure in the Co smological literature (see a gain [25]). As such, the inv estigation of their pro p er ties will fill a theoretical gap which is certainly o f int er e s t for CMB data analysis . Our aim in this pap er is then to investigate the correlatio n prop erties o f the Mexican needlets co e fficie nts. The sto chastic prop erties of w av elets ha ve bee n very extensively studied in the mathematical statistics literature, sta r ting from the seminal papers [10, 11]. W e m ust stress, how ever, that o ur fra mew or k here is very different: indeed, as expla ined ea r lier we shall b e co ncerned with 2 circumstances where obser v ations ar e made a t higher and higher fr e q uencies o n a single r ealization of a spherical rando m field. As such, no form of mixing or related pr op erties ca n b e assumed on the data and the proo fs will rely more directly on harmo nic metho ds, rather than on standard pro babilistic arg umen ts. W e shall pr ovide b oth a p ositive and a neg a tive result: namely , we will provide necessa ry a nd sufficient conditions for the Mexican needlets co efficients to b e uncorrelated, depending on the b ehaviour of the angula r pow er spectrum of the underlying (mean square contin uous and isotropic) random fields. In particular, on the contrary of what ha pp ens for the needlets in [2 6, 27], we shall show that there is indeed cor relation of the random co efficients whe n the angular power spe ctrum is decaying faster than a certain limit. Ho wev er, higher o rder versions (already co ns idered in [15]) of the Mexican needlets ca n indeed provide uncorrela ted co efficients, dep ending on a para meter which is r elated to the decay of the angular p ow er sp ectr um. In some sense, a heuristic ra tionale under these results can be expla ine d as follows: the co rrelation among co efficients is int ro duced basically by the presence in each of these terms of ra ndom elements which a re fixed (with resp ect to growing fre quencies) in a given realizatio n of the random field, b ecause they depend only on very larg e sca le b ehaviour (this is kno wn in the Physical literature as a Cosmic V ar iance effect). Becaus e of the compa ct supp ort in frequency in the ne e dlets as developed by [26, 2 7], these low-frequency comp onents are alwa ys dropp ed and unco rrelation is ensur ed. On the other hand, the same comp onents can b e dominant for Mexic a n needlets, in which ca ses it becomes necessary to in tro duce suitably mo dified v ersio ns which are b etter lo calized in the frequency domain (i.e., they allow less weigh t on very low frequency comp onents). As well-known, ther e is usually a trade-off b et ween lo caliza tion pro per ties in the frequency and real do mains, a s a consequence of the Uncer taint y Prin- ciple (“ I t is impo ssible for a non-zer o function a nd its F our ier trans fo rm to b e simult ane o suly very small”, see fo r instance [18]). In vie w of this, an interesting consequence o f our results can be loose ly sugges ted as follows: the b etter the lo calization in real doma in, the w or st the c orrelatio n pro p er ties . This is clearly a par a dox, a nd we do not try to formulate it more r igorous ly from the mathe- matical po in t of view - some numerical evidence will b e collected in an ongo ing, more applied work. Ho wev er, we ho pe that the previous dis c ussion will help to s he d some light within the class of needlets; in par ticula r, it should clarify that the uncorrelation pr op erty of w avelets co efficient s do es not follow at all by their lo calization prop erties in r eal domain. Indeed, g iven the fixed-domain asymptotics we a re consider ing, p erfect lo calization in real space do es not en- sure any form of unco rrelation (all random v alues at different lo cations on the sphere hav e in gener a l a non-zero correla tion). The plan of the pa p er is as follo ws: in Section 2 we s ha ll review so me basic results on isotropic random fields on the spher e a nd the (Mexican and stan- dard) needlets constructions. In Section 3 a nd 4 we establish our main results, providing necessary and s ufficien t conditio ns for the uncorrelation pro per ties to hold; in Section 5 we r eview some s tatistical applications. 3 2 Isotropic Random F ields and Spherical Needlets 2.1 Spherical R andom Fields In this paper, w e sha ll alwa ys b e concerned with zero-mea n, finite v ariance a nd isotropic random fields on the spher e, for which the following sp ectral represen- tation holds, in the mean squa r e sense : T ( x ) = X lm a lm Y lm ( x ) , x ∈ S 2 , (1) where { a lm } l,m , m = − l , ..., l is a tria ng ular ar r ay of zero-mean, orthogo nal, complex-v alued (for m 6 = 0) rando m v ariables with v ariance E | a lm | 2 = C l , the angular power s p ectrum o f the random field. The functions { Y lm ( x ) } are the so-called spherical ha rmonics, i.e. the eigenv ectors o f the Laplac ia n op erator o n the sphere [12], [34], [32] ∆ S 2 Y lm ( ϑ, ϕ ) =  1 sin ϑ ∂ ∂ ϑ  sin ϑ ∂ ∂ ϑ  + 1 sin 2 ϑ ∂ 2 ∂ ϕ 2  Y lm ( ϑ, ϕ ) = − l ( l + 1) Y lm ( ϑ, ϕ ) where w e have mov ed to spherical co or dinates x = ( ϑ , ϕ ), 0 ≤ ϑ ≤ π and 0 ≤ ϕ < 2 π . It is a well-known res ult that the spheric a l harmonics pro - vide a complete o r thonormal systems for L 2 ( S 2 ) . Ther e are many routes for establishing (1), usually by means of K arhunen-Lo ´ eve arg umen ts, the Sp ectral Representation Theorem, or the Sto ch as tic Peter-W e y l theorem, see for instance [1]. The spherical harmo nic co efficients a lm can b e recovered by means of the F ourier in version fo r mu la a lm = Z S 2 T ( x ) Y lm ( x ) dx . (2) If the random field is mean-s quare contin uous, the angular p ow er spe ctrum { C l } m ust satisfy the summability condition X l (2 l + 1) C l < ∞ . W e shall in tro duce a sligh tly stronger condition, as follo ws (see [3, 4, 13, 2 1]). Condition 1 F or al l B > 1 , ther e ex ist α > 2 , and { g j ( . ) } j =1 , 2 ,... a se quenc e of functions such that C l = l − α g j ( l B j ) > 0 , for B j − 1 < l < B j +1 , j = 1 , 2 , ... (3) wher e c − 1 0 ≤ g j ≤ c 0 for al l j ∈ N , and sup j sup B − 1 ≤ u ≤ B | d r du r g j ( u ) | ≤ c r , some c 0 , c 1 , ...c M > 0 , M ∈ N . 4 In practice, random fields such a s CMB are not fully observed, i.e. there are some missing observ ations in some regio ns of S 2 ; (2) is th us unfeasible in its exa ct form, and this motiv ates the introduction of spher ical wa velets such as needlets. 2.2 NPW Needlets The constructio n of the standar d needlet system is detailed in [2 6, 27]; we can lab el this system NPW ne e d lets and we sketc h here a few deta ils for complete- ness. Let φ b e a C ∞ function s uppo r ted in | ξ | ≤ 1, suc h that 0 ≤ φ ( ξ ) ≤ 1 and φ ( ξ ) = 1 if | ξ | ≤ 1 /B , B > 1. Define b 2 ( ξ ) = φ ( ξ B ) − φ ( ξ ) ≥ 0 so that ∀| ξ | ≥ 1 , ∞ X j =0 b 2 ( ξ B j ) = 1 . (4) It is immediate to v erify that b ( ξ ) 6 = 0 only if 1 B ≤ | ξ | ≤ B . The needlets frame  ϕ j k ( x )  is then constructed as ϕ j k ( x ) := p λ j k X l b ( l B j ) l X m = − l Y lm ( ξ j k ) Y lm ( x ) . (5) Here, { λ j k } is a set of cub atur e weights corr esp onding to the cub atur e p oi nt s  ξ j k  ; they ar e such to ensure that, for all p olynomia ls Q l ( x ) of degree smaller than B j +1 X k Q l ( ξ j k ) λ j k = Z S 2 Q l ( x ) dx . The main lo calization pr op erty of  ϕ j k ( x )  is established in [26], where it is shown that for a ny M ∈ N there exists a constant c M > 0 s.t., for every ξ ∈ S 2 :   ϕ j k ( ξ )   ≤ c M B j (1 + B j arccos h ξ j k , ξ i ) M uniformly in ( j, k ) . More explicitly , needlets are almost exp onentially lo calized aro und any c uba- ture p oint, which motiv ates their name. In the sto chastic ca se, the (random) spherical needlet coefficients are then defined as β j k = Z S 2 T ( x ) ϕ j k ( x ) dx = p λ j k X l b ( l B j ) l X m = − l a lm Y lm ( ξ j k ) . (6) It is then immediate to derive the cor r elation c o efficien t C or r  β j k , β j k ′  = E β j k β j k ′ q E β 2 j k E β 2 j k ′ = p λ j k λ j k ′ P l ≥ 1 b 2 ( l B j ) 2 l +1 4 π C l P l  ξ j k , ξ j k ′  p λ j,k λ j,k ′ P l ≥ 1 b 2 ( l B j ) 2 l +1 4 π C l . 5 where P l is the ultraspherica l (or Le g endre) po lynomial of order 1 2 and deg ree l ; the las t step follows fro m the w ell-known identit y [34] L l ( h ξ , η i ) := l X m = − l Y lm ( ξ ) Y lm ( η ) = 2 l + 1 4 π P l ( h ξ , η i ) . The ca pital sto chastic pr op erty for ra ndo m needlet co efficients is provided by [3], where it is shown that under Condition 1 the follo wing inequality ho lds   C or r  β j k , β j k ′    ≤ C M  1 + B j d  ξ j k , ξ j k ′  M , some C M > 0 , (7) where d  ξ j k , ξ j k ′  = arc c os  ξ j k , ξ j k ′  is the standard distance on the sphere. 2.3 Mexican N eedlets The constructio n in [1 5] is in a s ense s imila r to NPW needlets in [26, 2 7] (see also [14]), ins o far as a co m binatio n of L egendre po lynomials with a smo oth function is pro po sed; the main differ e nc e is that for NPW needlets the kernel is taken to b e compa ctly supp orted, whic h allows on o ne hand for an e x act r econ- struction function (needlets mak e up a tight fra me), at the same time granting exact lo calization in the frequency-domain. It should b e added, howev er, that the approa ch by [15] enjoys some undeniable strong p oints: firs tly , it cov ers general o riented manifolds and not simply the sphere; moreover it yields Gaus- sian loca lization pr op erties in the r eal do main. A further nice benefit is that it can be formulated in terms of a n explicit recip e in r eal s pace, a feature which is c e r tainly v aluable for practitioners. In pa r ticular, as we r epo rt b elow in the high-frequency limit the Mexican needlets are a symptotically close to the Spher- ical Mexica n Hat W av elets, which ha ve been exploited in several Cosmologica l pap ers but still la ck a s ound stochastic in vestigation. More precisely , [15] prop ose to replace b ( l/ B j ) in (5) b y f ( l ( l +1 ) /B 2 j ) , wher e f ( . ) is some Sch wartz function v anishing at zero (not neces s arily of b ounded suppo rt) a nd the se q uence { − l ( l + 1) } l =1 , 2 ,... represents the eigenv alues of the Laplacian op erator ∆ S 2 . In par ticular, Mexican needlets ca n b e obtained by taking f ( s ) = s ex p( − s ) , so to obta in ψ j k ;1 ( x ) = p λ j k X l ≥ 1 l ( l + 1) B 2 j e − l ( l +1) B 2 j L l  x, ξ j k  . The r esulting functions make up a fra me which is not tight, but very close to, in a sense which is ma de rigo rous in [15]. Exact cubature formulae ca nnot hold (in par ticular, { λ j k } ar e not exactly cuba ture weigh ts in this ca se), be c ause po lynomials of infinitely large o rder are in volved in the construction, but again this en tails very minor appr oximations in practical terms. More g enerally , it is poss ible to consider higher or der Mexican needlets b y fo cussing on f ( s ) = 6 s p exp( − s ) , so to obtain ψ j k ; p ( x ) := p λ j k X l ≥ 1 ( l ( l + 1 ) B 2 j ) p e − l ( l +1) /B 2 j L l  x, ξ j k  . The r andom spheric a l Mexican needlet co efficie nts are immediately seen to b e given by β j k ; p = Z S 2 T ( x ) ψ j k ( x ) dx = Z S 2 X l ≥ 0 l X m = − l a lm Y lm ( x ) ψ j k ; p ( x ) dx = p λ j k X l ≥ 1 ( l ( l + 1 ) B 2 j ) p e − l ( l +1) /B 2 j l X m = − l a lm Y lm  ξ j k  , whence their co v ariance is E β j k ; p β j k ′ ; p = p λ j k λ j k ′ X l ≥ 1 ( l ( l + 1 ) B 2 j ) p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l P l  ξ j k , ξ j k ′  . Throughout this pap er, we shall only consider weigh t functions of the form f ( s ) = s p exp( − s ) . It is ce r tainly p oss ible to cons ider more g eneral constr uc- tions; how ever this s pecific shape lends itself to very neat results, allowing us to pro duce b o th upp er a nd low er b ounds for the co efficients’ corr elation. Also, it makes p os sible a clea r interpretation o f the final results, i.e. the effect o f v ary ing p o n the structure of dep endence is immediately understo o d; this is, we b elieve, a v aluable a sset for practitioners . In the sequel, we shall dro p the subscript p , whenever p ossible without risk of confusion. Remark 2 As ment ione d e arlier, it is su ggeste d fr om r esult s in ([15]) that Mex- ic an n e e d lets pr ovide asymptotic al ly a very go o d appr oximation to the widely p opular Spheric al Mexic an Hat Wavelets (SMHW ) , which have b e en use d in many physic al p ap ers; the asymptotic analysis of the sto chastic pr op erties of SMHW c o efficients is stil l c ompletely op en for r ese ar ch. The discr etize d form of the SMHW c an b e written as Ψ j k ( θ ; B − j ) = 1 (2 π ) 1 2 √ 2 B − j (1 + B − 2 j + B − 4 j ) 1 2 [1 + ( y 2 ) 2 ] 2 [2 − y 2 2 t 2 ] e − y 2 / 4 B − j 2 , wher e the c o or di nates y = 2 tan θ 2 fol lows fr om the ster e o gr aphic pr oje ction on the tangent plane in e ach p oint of the spher e; her e we take θ = θ j k ( x ) := d ( x, ξ j k ) . Now write ψ j k ; p ( θ j k ( x )) = ψ j k ; p ( θ ) ; by fol lowing the ar gument s in [15 ] and developing t heir b ounds fu rt her, it c an b e ar gue d that   Ψ j k ( θ ; B − j ) − K j k ψ j k ; p ( θ )   = B − j O  min  θ 4 B 4 j , 1  , (8) 7 for some suitable n ormalization c onstant K j k > 0 . Equation (8) suggests t hat our r esults b elow c an b e use d as a guidanc e for the asymptotic the ory of r andom SMHW c o efficients. The validity of this appr oximation over r elevant c osmolo g- ic al mo dels and its implic ations for statistic al pr o c e dure s of CMB data analysis ar e curr ently b eing investigate d. 3 Sto c hastic pr op erties of Mexican needlet co- efficien ts, I: upp er b oun d s As men tioned in the Int ro ductio n, having established (7) op ened the wa y to several development s for the statistical analys is of spherical random fields. It is therefore a very impo rtant question to es ta blish under what circumstances these results can be extended to other constructions, such as Mexican needlets. In this a nd the following Section, we provide a full characteriza tion with p o sitive and negative r esults. W e sta rt by writing the expression for the correla tio n co efficients, which is given by C or r  β j 1 k 1 , β j 2 k 2  = P l ≥ 1 ( l ( l +1) B j 1 ) p ( l ( l +1) B j 2 ) p e − l ( l +1)( B − 2 j 1 + B − 2 j 2 ) (2 l + 1) C l P l  ξ j 1 k 1 , ξ j 2 k 2  ( P l ≥ 1 ( l ( l +1) B j 1 ) 4 p e − 2 l ( l +1) /B 2 j 1 (2 l + 1) C l ) 1 / 2 ( P l ≥ 1 ( l ( l +1) B j 2 ) 4 p e − 2 l ( l +1) /B 2 j 2 (2 l + 1) C l ) 1 / 2 . W e now provide upp er b ounds on the co rrelation of r andom coe fficien ts, as follows 1 . Theorem 3 Assum e Condition 1 holds with α < 4 p + 2 and M ≥ 4 p + 2 − α ; then ther e ex ist some c onstant C M > 0   C or r  β j 1 k 1 ; p , β j 2 k 2 ; p    ≤ C M  1 + B ( j 1 + j 2 ) / 2 − log B ( j 1 + j 2 ) / 2 d  ξ j k , ξ j k ′  (4 p +2 − α ) . (9) Pro of. W e pro ve (9) follo wing some idea s in [26]. F or notational simplicity , w e fo cus first on the case where j 1 = j 2 ; w e hav e C or r  β j k ; p , β j k ′ ; p  = P l ≥ 1 ( l ( l +1) B 2 j ) 2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l P l  ξ j k , ξ j k ′  P l ≥ 1 ( l ( l +1) B 2 j ) 2 p e − 2 l 2 /B 2 j 2 l +1 4 π C l . (10) 1 While finishing this pap er, we learned by p ersonal comm unication that wo rki ng indep en- den tly and at the same time as us, A.May eli has obt ained a result si milar to Theorem 3 for the case j 1 = j 2 , see [ 24 ]. The statemen ts and the assumptions in the t wo approaches are not equiv alen t and the methods of proofs are entirely differen t; w e b elieve b oth are of indep endent int erest. 8 Now replace C l = l − α g j  l B j  in the denominator o f the ab ov e repres en tation, for whic h w e get c − 1 0 B − (4 p +2 − α ) j 4 p + 2 − α ≤ X l ≥ 1 ( l ( l + 1) B 2 j ) 2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l ≤ c 0 B − (4 p +2 − α ) j 4 p + 2 − α . Denoting θ = arcco s  ξ j,k , ξ j,k ′  , the n umerator can be written as X l ≥ 1 ( l ( l + 1) B 2 j ) 2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l 1 π Z π θ sin  l + 1 2  ϕ (cos θ − cos ϕ ) 1 / 2 dϕ (11) where we hav e used the Dirichlet-Mehler integral r epresentation for the Legendre po lynomials [16]. The following steps and notations are v ery close to [2 6]. W e write C B ,g j = X l ≥ 1 ( l ( l + 1) B 2 j ) 2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π l − α g j  l B j  sin  l + 1 2  ϕ (12) : = 1 2 i X l ≥ 1 ( h j + ( l ) − h j − ( l )) where h j ± ( u ) = ( u ( u + 1) B 2 j ) 2 p 2 u + 1 4 π u − α g j  u B j  e − 2 u ( u +1) /B 2 j ± i ( u + 1 2 ) ϕ . By P ois son summatio n for m ula, we hav e X l ≥ 1 h j ± ( l ) = 1 2 X l ∈ Z h j ± ( l ) = 1 2 X µ ∈ Z b h j ± (2 π µ ) . Denote G α,j ( t ) := t 2 p − α g j ( t ) e − 2 t ( t + B − j ) I ( B − j , ∞ ) . (13) Let us no w recall the follo wing standard prop erty of F ourier transforms: ( iω ) k d m dω m b f ( ω ) = ( d k dx k { x m f ( x ) } b )( ω ) . Some simple computations yield b h ± (2 π µ ) = B (1 − α ) j 4 π ( 2 p X m =1 [2  2 p m − 1  +  2 p m  ] B − (2 p +1 − m ) j d m dω m + B − (2 p +1) j + 2 d 2 p +1 dω 2 p +1 ) × Z ∞ −∞ G α,j  t/B j  e ± i ( t + 1 2 ) ϕ − itω dt | ω =2 πµ = B (2 − α ) j e ± iϕ 4 π ( 2 p X m =1 [2  2 p m − 1  +  2 p m  ] B − (2 p +1 − m ) j d m dω m ) b G α,j ( ω )   ω = B j (2 π µ ∓ ϕ ) + B (2 − α ) j e ± iϕ 4 π  B − (2 p +1) j + 2 d 2 p +1 dω 2 p +1  b G α,j ( ω )   ω = B j (2 π µ ∓ ϕ ) , 9 where b G α,j ( ω ) = Z R G α,j ( t ) e − itω dt . F or all positive in tegers k ≤ M , we can obtain     Z ∞ B − j d k dt k { t m G α,j ( t ) } dt     ≤ ( k Γ( m +2 p − α ) C g L ( p,m,α,k ) B − j (2 p +1+ m − α − k ) , for 2 p + 1 − α + m 6 = k k Γ ( m + 2 p − α ) C g ( j lo g B ) , for 2 p + 1 − α + m = k (14) where C g = max { c 0 , ..., c M } and L ( p, m , α, k ) :=  Γ (2 p + 1 + m − α − k ) when (2 p + 1 + m − α − k ) > 0 , (Γ ( k − 2 p − 1 − m + α )) − 1 when (2 p + 1 + m − α − k ) < 0 . It should b e noticed that o ur argument here differs from the one in [26], b e cause we cannot as s ume the integrand function on the left-hand side to b e in L 1 for all k ≤ M . Let us now fo cus on the case where k = 4 p + 2 − α, with m = 2 p + 1; we obtain     d 2 p +1 dω 2 p +1 b G α,j ( ω )       B j (2 π µ − ϕ )   4 p +2 − α ≤     Z ∞ B − j d k dt k  t 2 p +1 G j ( t )  dt     ≤ (4 p + 2 − α ) Γ (4 p + 1 − α ) C g ( j lo g B ); therefore     d 2 p +1 dω 2 p +1 b G α,j ( ω )     ≤ C 2 p +1 ,α,g B − j (4 p +2 − α )+log B j (2 π µ − ϕ ) 4 p +2 − α , where C 2 p +1 ,α,g = (4 p + 2 − α ) Γ (4 p + 1 − α ) C g log B . The modifications needed for other cases are o b vio us and we o bta in B − (2 p +1 − m ) j     d m dω m b G α,j ( ω )     ≤ C m,α,g B − j (4 p +2 − α ) (2 π µ − ϕ ) 4 p +2 − α , where C m,α,g = (4 p + 2 − α ) Γ ( m + 2 p − α ) C g L ( p, m , α, k ) . Now let C α,g = max { C m,α,g , m = 0 , ..., 2 p + 1 } ; we have    b h ± (2 π µ )    ≤ B (2 − α ) j 4 π C α,g ( 2 p X m =0  2 p m  B − j (4 p +2 − α ) (2 π µ − ϕ ) 4 p +2 − α + B − j (4 p +2 − α )+log B j (2 π µ − ϕ ) 4 p +2 − α ) ≤ 2 2 p C α,g B − 4 pj +log B j (2 π µ ∓ ϕ ) 4 p +2 − α , µ = 1 , 2 , ... (15) 10 Therefore   C B ,g j   ≤ 2 2 p C α,g   1 2 ϕ 4 p +2 − α + X µ ∈ N 1 | 2 π µ ± ϕ | 4 p +2 − α   B − 4 pj +log B j ≤ 2 2 p  1 2 ϕ 4 p +2 − α + | 4 p + 1 − α | π α − 4 p − 1  C α,g B − 4 pj +log B j . Hence, for the n umerator of the co rrelation w e hav e the b ound (11) ≤ C B − 4 pj +log B j Z π θ  1 2 ϕ 4 p +2 − α + | 4 p + 1 − α | π α − 4 p − 1  (cos θ − cos ϕ ) 1 / 2 dϕ . As in [2 6], when 0 ≤ θ ≤ π / 2 , we can get the following inequality (11) ≤ C α,k, g B − 4 pj +log B j Z π θ π α − 4 p − 1 0 . 27 ϕ 4 p +2 − α  ϕ 2 − θ 2  1 / 2 dϕ ≤ C 1 B − 4 pj +log B j θ α − 4 p − 2 . If π / 2 ≤ θ ≤ π , letting e θ = π − θ, e ϕ = π − ϕ, we can obtain the same b ound. Going back to (10), we obtain C or r  β j,k , β j,k ′  ≤ C θ α − 4 p − 2 B − 4 pj +log B j B (2 − α ) j ≤ C θ α − 4 p − 2 B − ( j − log B j )(4 p +2 − α ) → 0 , as j → ∞ , W e th us get inequality (9) for j 1 = j 2 . Now let us consider j 1 6 = j 2 . As the pro o f is very similar to the arguments ab ov e, we omit many details. In the seq uel, for an y t wo sequences a l , b l , w e write a l ≈ b l if and o nly if a l = O ( b l ) a nd b l = O ( a l ) . First, w e co nsider the v ariance o f the random co efficients, which can b e represented by: X ( l ( l + 1) B j 1 ) 4 p e − 2 l ( l +1) /B 2 j 1 (2 l +1) C l ≈ B (2 − α ) j 1  Z 1 B − j 1 + Z ∞ 1  t 4 p +1 − α e − 2 t ( t + B − j 1 ) g j 1 ( t ) dt  = B (2 − α ) j 1  c 0 4 p + 2 − α B − j 1 (4 p +2 − α ) + O (1)  ; therefore    X l ≥ 1 ( l ( l + 1 ) B j 1 ) 4 p e − 2 l ( l +1) /B 2 j 1 (2 l + 1) C l    1 / 2    X l ≥ 1 ( l ( l + 1 ) B j 2 ) 4 p e − 2 l ( l +1) /B 2 j 2 (2 l + 1) C l    1 / 2 = B (1 − α/ 2)( j 1 + j 2 )  c 0 4 p + 2 − α B − j 1 (4 p +2 − α ) + O (1)  1 / 2  c 0 4 p + 2 − α B j 2 (4 p +2 − α ) + O (1)  1 / 2 = O (1 ) B (1 − α/ 2)( j 1 + j 2 ) . 11 Without loss of g enerality , we ca n always assume j 1 < j 2 . W e c a n implement the same argument a s b efore, provided we r eplace C B ,g j in (12) b y C B ,g ,j 1 ,j 2 = X l ≥ 1 ( l ( l + 1) B j 1 + j 2 ) 2 p e − l ( l +1)( B − 2 j 1 + B − 2 j 2 ) 2 l + 1 4 π l − α g j 1  l B j 1  sin  l + 1 2  ϕ = : 1 2 i X l ≥ 1 ( h j 1 j 2 + ( l ) − h j 1 j 2 − ( l )) where h j 1 j 2 ± ( u ) = ( u ( u + 1) B j 1 + j 2 ) 2 p 2 u + 1 4 π u − α g j 1  u B j 1  e − u ( u +1)( B − 2 j 1 + B − 2 j 2 ) ± i ( u + 1 2 ) ϕ . Again, b y Poisson summation formula, we have X l ≥ 1 h j 1 j 2 ± ( l ) = 1 2 X l ∈ Z h j 1 j 2 ± ( l ) = 1 2 X µ ∈ Z b h j 1 j 2 ± (2 π µ ) . Denote G α,j 1 j 2 ( t ) := t 2 p − α g j 1 ( t ) e − t ( t + B − j 1 )( 1+ B 2( j 1 − j 2 ) ) I ( B − j 2 , ∞ ) ; by the same argument and notation as in (1 5), we have    b h j 1 j 2 ± (2 π µ )    = B 2 p ( j 1 − j 2 )+(2 − α ) j 1 4 π ( 2 p X m =1 [2  2 p m − 1  +  2 p m  ] B − (2 p +1 − m ) j 1 d m dω m ) b G α,j 1 j 2 ( ω )   ω = B j 1 (2 π µ ∓ ϕ ) + B 2 p ( j 1 − j 2 )+(2 − α ) j 1 e ± iϕ 4 π  B − (2 p +1) j 1 + 2 d 2 p +1 dω 2 p +1  b G α,j 1 j 2 ( ω )   ω = B j 1 (2 π µ ∓ ϕ ) ≤ 2 2 p C α,g B − 2 p ( j 1 + j 2 )+log B j 1 (2 π µ ∓ ϕ ) 4 p +2 − α , µ = 1 , 2 , ... Therefore   C B ,g j 1 ,j 2   ≤ 2 2 p C α,g  1 2 ϕ 4 p +2 − α + | 4 p + 1 − α | π α − 4 p − 1  B − 2 p ( j 1 + j 2 )+log B j 1 . It is then stra ightf o r ward to conclude as in the c a se where j 1 = j 2 , to obtain C or r  β j 1 ,k , β j 2 ,k ′  ≤ C θ α − 4 p − 2 B − 2 p ( j 1 + j 2 )+log B j 1 B (1 − α/ 2)( j 1 + j 2 ) ≤ C θ α − 4 p − 2 B − [( j 1 + j 2 ) / 2 − log B ( j 1 + j 2 ) / 2](4 p +2 − α ) → 0 , as j 2 → ∞ . Thu s (9) is established. 12 Remark 4 By c ar eful manipulation, we c an obtain an explicit expr ession for the c onstant C M in (9 ), i.e. C M = 2 2 p π M +1 (4 p + 2 − α ) 2 Γ (4 p + 1 − α ) c 0 C g log B . The previous result sho ws that Mexican needlets can enjoy the s ame uncor - relation prop erties as standar d ne e dlets, in the circumstances where the ang ular power sp ectrum is decaying “s lowly enough”. The extr a log ter m in (9) is a con- sequence of a standard technical difficulty whe n dealing with a b oundary ca se in the in tegra l in (14). Remark 5 T o obtain c entr al limit r esults for fin ite-dimensional statistics b ase d on nonline ar tr ansformations of the Mexic an ne e d lets c o efficients, it would b e sufficient to c onsider the c ase wher e j = j ′ . The asymptotic unc orr elation we establishe d in The or em 3 is str onger; inde e d, for many applic atio ns it is useful to c onsider differ ent sc ales { j } at t he same time. Be c ause of this, it is also imp or- tant to fo cus on the c orr elatio n of Mexic an ne e d let c o efficients at differ ent j, j ′ . We str ess that the ne e d for su ch analysis was mu ch mor e limite d for standar d ne e d lets; inde e d in the latter c ase, given the c omp actly supp orte d kernel b ( . ) , the fr e quency supp ort of the various c o efficients is aut omatic ally disjo int when | j − j ′ | ≥ 2 , whenc e (for c ompletely observe d r andom fields) st andar d ne e d let c o effici ent s c an b e c orr elate d only for | j − j ′ | = 1 . 4 Sto c hastic pr op erties of Mexican needlet co- efficien ts, I I: lo w er b ou n ds In this Section, we complete the prev ious analys is , establishing indeed that the random Mexic an needlets co efficients a re necessa rily correla ted at some angula r distance in the pres e nce of faster memo ry decay . This is clearly differ e n t from needlets, which a re always unco rrelated. As mentioned in the Introduction, the heuristic r ationale b ehind this duality can b e explained as follo ws: it should b e stressed that we are fo cussing on high-r e s olution asy mptotics, i.e. the asymp- totic b ehaviour o f rando m co efficient s a t smaller and smaller scales in the same random r ealization. F or such asymptotics, a c r ucial ro le can be played by terms which remain constant acros s differen t s cales. In the ca se of us ual needlets, which have bo unded s uppor t ov er the m ultipo les, terms like thes e are simply dropp ed by co ns truction. This is not so for Mexica n needlets, which in any case include comp onents a t the lo west scales. These co mponents are dominant when the angular p ower sp ectrum decays fas t, a nd a s such they prevent the po ssibility of asymptotic uncor relation. In par ticular, we hav e corr elation when the angular pow er spec tr um is such that α > 4 p + 2 . 13 Theorem 6 Under c ondition 1, for α > 4 p + 2 , ∀ ε ∈ (0 , 1) , t her e exists a p ositive δ ≤ ε  1 + c 2 0  − 1 / ( α − 4 p − 2) such that lim j →∞ inf C or r  β j k ; p , β j k ′ ; p  > 1 − ε , (16) for al l  ξ j k , ξ j k ′  such t hat d ( ξ j k , ξ j k ′ ) ≤ δ . Pro of. W e first divide the v aria nce of the co efficients into three pa rts, as follows   X 1 ≤ l<ǫ 1 B j + X ǫ 1 B j ≤ l<ǫ 2 B j + X l ≥ ǫ 2 B j    l ( l + 1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l = : A 1 j + A 2 j + A 3 j . Int uitively , our idea is to sho w that the first sum is of exact order O ( B (2 − α ) j × B ( α − 4 p − 2) j ) = O ( B − 4 pj ) , while the second t wo are smaller ( O ( B (2 − α ) j ) and o ( B (2 − α ) j ), resp ectively). Indeed, for the first part we obtain easily A 1 j = X 1 ≤ l<ǫ 1 B j  l ( l + 1) B 2 j  2 p e − 2 l ( l − 1) /B 2 j 2 l + 1 4 π C l ≤ 2 X 1 ≤ l<ǫ 1 B j l 4 p B 4 pj l π l − α g j ( l B j ) ≤ 2 c 0 π B (2 − α ) j Z ǫ 1 B − j x 4 p +1 − α dx = 2 c 0  B ( α − 4 p − 2) j − ǫ 4 p +2 − α 1  π ( α − 4 p − 2) B (2 − α ) j , and X 1 ≤ l<ǫ 1 B j  l ( l + 1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l ≥  B ( α − 4 p − 2) j − ǫ 4 p +2 − α 1  2 π c 0 ( α − 4 p − 2) e − 2 ǫ 2 1 B (2 − α ) j . Similarly , for the second part  ǫ 4 p +2 − α 2 − ǫ 4 p +2 − α 1  2 π ( α − 4 p − 2) c 0 e − 2 ǫ 2 2 B (2 − α ) j ≤ X ǫ 1 B j ≤ l<ǫ 2 B j  l ( l + 1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l ≤ 2  ǫ 4 p +2 − α 1 − ǫ 4 p +2 − α 2  π ( α − 4 p − 2) c 0 e − 2 ǫ 2 1 B (2 − α ) j , and for the third par t X l ≥ ǫ 2 B j  l ( l + 1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l ≤ 3 c 0 4 π B (2 − α ) j Z ∞ ǫ 2 x 4 p +1 − α e − x 2 dx ≤ 3 c 0 4 π B (2 − α ) j ǫ 4 p +1 − α 2 Z ∞ ǫ 2 e − x 2 dx ≤ 3 c 0 4 π ǫ 4 p − α 2 e − ǫ 2 2 B (2 − α ) j , ( ǫ 2 > 1) . 14 The last inequalit y follows from the asymptotic formula Z ∞ y e − x 2 / 2 dx ∼ 1 y e − y 2 / 2 , y → ∞ . W e hav e then that A 3 j A 1 j =    X 1 ≤ l<ǫ 1 B j  l ( l + 1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l    − 1 ×    X l ≥ ǫ 2 B j  l ( l + 1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l    = O ( B j (4 p +2 − α ) ) = o (1) , as j → ∞ . On the other hand, for any po sitive ε < 1 , if w e c ho ose ǫ 1 = N B − j , where N is sufficiently large that N 4 p +2 − α  1 + 2 ε c 2 0  < 1 , we obtain that  B ( α − 4 p − 2) j − ǫ 4 p +2 − α 1  c 0 > 2 ε  ǫ 4 p +2 − α 1 − ǫ 4 p +2 − α 2  c 0 , whence A 2 j A 1 j < ε 2 as j → ∞ . Thu s we obtain that    X l ≥ 1  l ( l + 1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l + 1 4 π C l    − 1 = 1 A 1 j  1 + A 2 j A 1 j + A 3 j A 1 j  − 1 ≥ 1 A 1 j n 1 + ε 2 + o (1) o − 1 ≥ cB − 4 pj , some c > 0 . More explicitly , the v ariance at the denominator has the same order as for the summation restricted to the elemen ts in the r ange 1 ≤ l < ǫ 1 B j . T o analyze the n umerator , we sta rt b y recalling that sup θ ∈ [0 , π ] P l (cos θ ) = P l (cos 0) = 1 , and sup θ ∈ [0 , π ] | d dθ P l (cos θ ) | ≤ 3 l . As a conseq uenc e , for any ε > 0 there exists a δ > 0 , s.t. if 0 < θ ≤ δ ≤ ε/ 6 N , then, | P l (cos θ ) − P l (cos 0) | ≤ 3 l θ ≤ ε , for all l > N , where the ab ov e ineq ualities follow from 0 ≤ c o s ( θ 0 + θ ) − co s θ 0 = 2 sin 2 θ 2 ≤ θ , ∀ θ 0 ∈ [0 , 2 π ) . 15 Therefore, for an y ξ j k , ξ j k ′ ∈ S 2 s.t. arcc os  ξ j k , ξ j k ′  ≤ δ , we hav e C or r  β j k , β j k ′  = P l ≥ 1  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l P l  ξ j k , ξ j k ′  P l ≥ 1  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l ≥ P 1 ≤ l ≤ N  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j C l P l (cos 0) P l ≥ 1  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l − P 1 ≤ l ≤ N  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l   P l  ξ j k , ξ j k ′  − P l (cos 0)   P l ≥ 1  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l + P l>N  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j C l   P l  ξ j k , ξ j k ′    P l ≥ 1  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l ≥ P 1 ≤ l ≤ N  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l × (1 − ε ) P l ≥ 1  l ( l +1) B 2 j  2 p e − 2 l ( l +1) /B 2 j 2 l +1 4 π C l + O ( B j (4 p +2 − α ) ) = (1 − ε ) 1 + ε/ 2 + o (1) = (1 − ε ′ ) + o (1 ) , as j → ∞ , ε ′ = 3 ε 2 + ε . Thu s the pro of is completed. As mentioned earlier, the res ults in the pr evious tw o Theorems illustrate an interesting trade-off b etw een the lo c alization and cor relation pro per ties of spherical needlets. In pa rticular, we can a lw ays achieve uncorrelatio n by cho os- ing p > ( α − 2) / 4; of course α is generally unkno wn and m ust be estimated from the data (in this sense standar d needlets hav e b etter robustness prop er- ties). Introducing hig he r order terms implies lowering the w eight of the lowest m ultip oles, i.e . impr oving the lo caliza tion prop erties in frequency s pa ce. On the other hand, it may b e expected that such a n impro vemen t o f the lo ca lization prop erties in the frequency do main will lea d to a worsening of the lo calization in pixel space, as a consequence of the Uncertaint y Principle we mentioned in the Int ro ductio n (see for instance [18]). W e do not inv estiga te this issue here, but we sha ll provide some n umerical e vidence on this pheno meno n in an ong oing work. 16 Remark 7 In The or em 6, we de cide d to ke ep t he assumptions as close as p os- sible to The or em 3, in or der to e ase c omp arisons and highligh t the symmetry b etwe en the two r esults. However, it is simple to show t hat the c orr elation r esult holds in much gr e ater gener ality, for angular p owe r sp e ctr a that have a de c ay which is faster than p olynomial. In p articular, assume that C l = H ( l ) e x p( − l p ) , l = 1 , 2 , ... wher e H ( l ) is any kind of p olynomial s uch t hat H ( l ) > c > 0 and p > 0 . Th en it is simple to establish the s ame r esult as in The or em 6, by m e ans of a simplifie d version of the same ar gument. The underlying r ationale should b e e asy to get: for exp onential ly de c ayi ng p ower sp e ctr a the dominating c omp onents ar e at the lowest fr e quencies, and they int r o duc e c orr elations among al l r andom c o effici ents which c annot b e ne gle cte d. 5 Statistical Applications The pr e vious res ults lend themselves to several a pplications for the statistica l analysis o f spherical r andom fields, in particular with a view to CMB data analysis. Simila rly to [3], let us consider p olynomials functions of the normaliz ed Mexican needlets co efficien ts, as follows h u,N j := 1 p N j N j X k =1 Q X q =1 w uq H q ( b β j k ; p ) , b β j k ; p := β j k ; p q E β 2 j k ; p , u = 1 , ..., U , where H q ( . ) deno tes the q - th order Hermite polyno mials (see [33]), N j is the cardinality of co efficients cor resp onding to fr equency j (where we take  ξ j k  to form a B − j -mesh, s ee [4], so that N j ≈ B 2 j ), and { w uq } is a se t of deter ministic weigh ts that m ust e ns ure these statistics ar e as y mptotically nondegener ate, i.e. Condition 8 Ther e exist j 0 such t hat for al l j > j 0 rank (Ω j ) = U , Ω j := E h N j h ′ N j , h N j := ( h 1 ,N j , ..., h U,N j ) ′ . Condition (8) is a standard inv ertibility ass umption which will ensure o ur statistics a re asymptotically nondegenera te (for instance, it rules o ut multi- collinearity). Several exa mples of relev ant po lynomials are given in [3 ]; for instance, given a theo retical mo del for the a ngular p ow er sp ectr um { C l } , it is 17 suggested in that re fer ence that a g o o dness-of-fit statistic migh t be based up on 1 p N j N j X k =1 H 2 ( b β j k ; p ) = 1 p N j N j X k =1 ( b β 2 j k ; p − 1) = 1 p N j N j X k =1 ( β 2 j k ; p λ j k P l ≥ 1 b 2 ( l B j ) 2 l +1 4 π C l − 1) . The statistic b Γ j = 1 N j N j X k =1 β 2 j k ; p λ j k can then be viewed as an un biased es tima to r for Γ j = E b Γ j = X l ≥ 1 b 2 ( l B j ) 2 l + 1 4 π C l . W e r efer to [13], [8] for the analysis of this e s timator in the pres ence of miss- ing o bs erv ations and noise, and for its application to CMB da ta in the stan- dard needlets case. Our r e sults b elow can b e viewed as providing co ns istency and as ymptotic Gaussianit y (for fully observed maps and without noise) in the Mexican needlets approach. As alwa ys in this framework, consis tency has a non-standar d meaning, as we do not hav e conv er gence to a fixed para meter, but rather con vergence to unity of the ratio b Γ j / Γ j . Likewise, tests of Gaussianity could b e implemented by fo cussing on the skewness and kurtosis of the wa velets co efficients (see for instance [25]), i.e. b y fo cussing on 1 p N j N j X k =1 n H 3 ( b β j k ; p ) + H 1 ( b β j k ; p ) o = 1 p N j N j X k =1 b β 3 j k ; p and 1 p N j N j X k =1 n H 4 ( b β j k ; p ) + 6 H 2 ( b β j k ; p ) o = 1 p N j N j X k =1 n b β 4 j k ; p − 3 o . The join t distribution for these statistics is pr ovided b y the following results: Theorem 9 Assum e T is a Gaussian me an squar e c ontinuous and isotr opic r andom fi eld; assume also that Conditions 1, 8 ar e satisfie d and cho ose p > ( α + δ ) / 4 , some δ > 0 . Then as N j → ∞ Ω − 1 / 2 j h N j → d N (0 , I U ) . Pro of. The asymptotic behaviour o f our polyno mial statis tics can be estab- lished by means o f the method of moments. In particular, it is p oss ible to 18 exploit the diagra m formula for higher order moments of Hermite p olyno mial, as explained for instance in [28],[3 3]. The details are the same as in [3], and th us they are o mitted for brevit y’s sake. W e only note that, in o rder to be able to use Lemma 6 in that reference, we need to ensure that   C or r  β j k ; p , β j k ′ ; p    ≤ C  1 + B j d  ξ j k , ξ j k ′  2+ δ , some C > 0 . In view o f (9), this motiv ates the tighter limit we need to imp ose on the v alue of p. It may b e noted that the cov ar iance matrix Ω j can itself b e consis ten tly estimated from the data at any level j, for ins tance by means of the b o ot- strap/subsa mpling techniques tha t are detailed in [4]. 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W or ld Scien tific, Singap or e [35] W iaux, Y., McEwen, J.D., Vielv a, P ., (20 07) Complex Data Pr o- cessing: F as t W av elet Analysis o n the Sphere, Journal of F ourier A nalysis and its Applic ations, 13, 477-494 21 Corresp onding Author: Domenico Marinucci Dipartimento di Matematica Univ er sita’ di Roma T o r V er gata via della Ricerca Scien tifica 1 00133 Italy email: mar inucc@mat.uniroma2 .it tel. +39 -06725 94105 fax +39-0 67259 4699 22