Probab. Theory Relat. Fields (2008) 141:389–409 DOI 10.1007/s00440-007-0088-8
A central limit theorem and higher order results for the angular bispectrum Domenico Marinucci
Received: 15 July 2005 / Revised: 16 May 2007 / Published online: 17 July 2007 © Springer-Verlag 2007
Abstract The angular bispectrum of spherical random fields has recently gained an enormous importance, especially in connection with statistical inference on cosmological data. In this paper, we analyze its moments and cumulants of arbitrary order and we use these results to establish a multivariate central limit theorem and higher order approximations. The results rely upon combinatorial methods from graph theory and a detailed investigation for the asymptotic behavior of coefficients arising in matrix representation theory for the group of rotations S O(3). Keywords Spherical random fields · Angular bispectrum · Central limit theorem · Higher order approximations Mathematics Subject Classification (2000) Primary: 60G60; Secondary: 60F05 · 62M15 · 62M40
1 Introduction Let T (θ, ϕ) be a random field indexed by the unit sphere S 2 , 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π . We assume that T (θ, ϕ) has zero mean, finite variance and it is mean square continuous and isotropic, i.e., its covariance is invariant with respect to the group of
I am very grateful to an associate editor and two referees for many useful comments, and to M. W. Baldoni and P. Baldi for discussions on an earlier version. D. Marinucci (B) Department of Mathematics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy e-mail:
[email protected] 123
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rotations S O(3); the following spectral representation holds in mean square sense [14]: ∞ l alm Ylm (θ, ϕ). (1) T (θ, ϕ) = l=1 m=−l
Here, we use {Ylm (θ, ϕ)} to denote the the spherical harmonics (for explicit expressions see [22, Chap. 5], [17]). The triangular array {alm } represents a set of random coefficients, which can be obtained from T (θ, ϕ) through the inversion formula π π alm =
∗ T (θ, ϕ)Ylm (θ, ϕ) sin θ dθ dϕ, m = 0, ±1, . . . , ±l, l = 1, 2, . . . ;
−π 0
(2) see for instance Kim, Koo and Park [11] for a review of Fourier analysis on S 2 . The coefficients alm are zero-mean and uncorrelated , with variance E|alm |2 = Cl ; ∗ ), for they are real-valued for m = 0, complex-valued (with al,−m = (−1)m alm m = ±1, . . . , ±l. Hence, if T (θ, ϕ) is Gaussian, they have a Gaussian distribution (complex-valued, in the sense of [5], for m = 0), and they are independent over l and m ≥ 0 (see [2] for a converse statement). The index l is usually labeled a multipole; approximately, each multipole corresponds to an angular resolution given by π/l. The sequence {Cl } denotes the angular power spectrum: we shall always assume that Cl is strictly positive, for all values of l. The interest on spherical random fields has recently gained momentum, due to strong empirical motivations arising especially (but not exclusively) from the statistical analysis of Cosmic Microwave Background radiation (CMB), see for instance Dodelson [7] for details. The first spherical maps of CMB data were released by the satellite experiment COBE in 1992; for these results G. Smoot and J. Mather were awarded the Nobel Prize for Physics in 2006. Much more detailed data have been collected by the NASA mission WMAP, which released its maps in 2003 and 2006; more refined observations are expected from the ESA mission Planck, which will be launched in 2008. Given these maps, by means of (2) it is possible to recover the coefficients {alm } for arbitrary large values of the multipole l. In practice, however, the values of l which can be achieved are limited by the technical features of a given experiment, in particular, the resolution of its detectors and their instrumental noise. For instance, this upper limit on l is around 20 for COBE data, approximately 600/700 for the experiment WMAP, and it is expected to be in the order of 2,500 for Planck. In other words, in each of these experiments the same realization of a single spherical random field is observed; more data in this framework simply means that higher and higher frequencies become available for statistical inference. The asymptotic theory developed in this paper reflects the features of this experimental environment; in particular, the limit theorems are derived with respect to increasing multipoles (i.e., frequencies) rather than an increasing span of observations. This approach can be interpreted as a form of fixed-domain asymptotics, a framework which has become quite popular in recent years, see for instance Stein [19] or Loh [15].
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Among a variety of statistical issues raised by CMB data analysis, several papers have focussed in particular on testing for non-Gaussianity by a variety of non-parametric methods (see [16] for a review). The majority of efforts has focussed on the bispectrum, which is considered an optimal statistic to verify the accuracy of the so-called inflationary scenario, the leading paradigm for the dynamics of the Big Bang [12,13]. For observations on stationary stochastic processes defined on Zk or Rk , k ≥ 1, the bispectrum is a well-known and deeply studied statistic, see for instance Brillinger [5], Subba Rao and Gabr [20]. In that framework, standard Fourier transforms can be implemented by means of complex exponentials and the bispectrum is defined to be invariant under the action of the corresponding Abelian group of translations. The circumstances considered in this paper are quite different; indeed, in the spherical case the (angular) bispectrum is defined to be invariant under the action of the group of rotations S O(3); its minimum mean square error estimator is provided by [10] l1
Bl1 l2 l3 :=
l2
l3 l1 m1
m 1 =−l1 m 2 =−l2 m 3 =−l3
l2 m2
l3 m3
(al1 m 1 al2 m 2 al3 m 3 ),
where l1 + l2 + l3 is constrained to be even. The statistic Bl1 l2 l3 is real-valued and it is invariant with respect to permutation of its arguments l1 , l2 , l3 . The “Wigner’s 3 j symbols” appearing on the right-hand side are defined by [22, Expression 8.2.1.5]
l1 l2 l3 m1 m2 m3
(l1 +l2 −l3 )!(l1 − l2 + l3 )!(l1 − l2 + l3 )! := (−1) (l1 + l2 + l3 + 1)! 1/2 (l3 + m 3 )!(l3 − m 3 )! × (l1 + m 1 )!(l1 − m 1 )!(l2 + m 2 )!(l2 − m 2 )! l3 +m 3 +l2 +m 2
×
z
1/2
(−1)z (l2 + l3 + m 1 − z)!(l1 − m 1 + z)! , z!(l2 + l3 − l1 − z)!(l3 + m 3 − z)!(l1 − l2 − m 3 + z)!
where the summation runs over all z’s such that the factorials are non-negative. Up to a constant factor, the Wigner’s 3 j symbols are equivalent to the so-called Clebsch– Gordan coefficients, which play a crucial role in group representation theory [23]. A normalized and phase-adjusted version of the angular bispectrum can be defined by Bl l l Il1 l2 l3 := (−1)(l1 +l2 +l3 )/2 1 2 3 . (3) Cl 1 Cl 2 Cl 3 In practice Il1 l2 l3 is unfeasible because Cl is unknown. In view of (2), and neglecting measurement errors, a feasible estimator for Cl is l := C
l 1 |alm |2 , l = 1, 2, . . . , 2l + 1
(4)
m=−l
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which is clearly unbiased (see also [1]). Thus Il1 l2 l3 can be replaced by the feasible statistic Bl l l (5) Il1 l2 l3 := (−1)(l1 +l2 +l3 )/2 1 2 3 . l1 C l2 C l3 C Although the angular bispectrum has been the object of an enormous attention in the cosmological literature, very few analytic results are so far available on its probabilistic properties. In a previous paper [18], we established some bounds on the behavior of the first eight moments of (3)/(5), and we used these results to investigate the asymptotic behavior of some functionals of the bispectrum array; such functionals were proposed to build non-parametric tests for non-Gaussianity, and these tests were shown to be very effective for the analysis of CMB data [6]. These asymptotic results were established by means of a direct analysis of cross-products of Wigner’s 3 j coefficients; the analysis was performed by means of explicit summation formulae originated from the quantum angular momentum literature (see [22] for a very detailed collection of results). In the present paper, these results are made sharper and extended to moments and cumulants of arbitrary orders by means of a more general argument. More precisely, we show how it is possible to associate to higher order cumulants some algebraic coefficients arising in matrix representation theory for the group of rotations S O(3). We are then able to obtain combinatorial expressions for moments and cumulants of arbitrary orders; these results are exploited to obtain multivariate central limit theorems and higher order approximations.
2 A central limit theorem for the bispectrum We shall establish the Central Limit Theorem by means of the methods of cumulants. As well-known, cumulants (or semi-invariants) are defined as the coefficients in the Taylor expansion of the log-characteristic function; see for instance Brillinger [5], Surgailis [21] for explicit expressions and properties, for instance their (one-to-one) relationship to higher order moments. Let X 1 , . . . , X I be complex-valued random variables; we shall denote their joint cumulants by χ p1 p2 .... p I (X 1 , X 2 , . . . , X I ). As well known χ1 (X ) = EX and χ11 (X 1 , X 2 ) = Cov(X 1 , X 2 ). Concerning the bispectrum (3), it is a standard result that the random variables
angular in the array Il1 l2 l3 have mean zero, variance EIl21 l2 l3 = l1 l2 l3 and are uncorrelated for all triples (l1 , l2 , l3 ) = (l1 , l2 , l3 ) : we recall that
l1 l2 l3 := 1 + δll12 + δll23 + 3δll13 =
123
⎧ ⎨
1 for l1 < l2 < l3 2 for l1 = l2 < l3 or l1 < l2 = l3 ; ⎩ 6 for l1 = l2 = l3
A central limit theorem and higher order results for the angular bispectrum
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here and in the sequel, δab denotes Kronecker’s delta, that is δab = 1 for a = b, zero otherwise. It is also easily seen that χ111 (Il11 l12 l13 , Il21 l22 l23 , Il31 l32 l33 ) ≡ 0, for all l11 , . . . , l33 . The following Proposition extends these results providing a bound for joint cumulants of arbitrary order. In the sequel of this paper we shall always rearrange terms so that li1 ≤ li2 ≤ li3 for all i and li1 ≤ li 1 when i ≤ i . Theorem 2.1 Assume that (li1 , li2 , li3 ) = (li 1 , li 2 , li 3 ) whenever i = i , where I ≥ 1 and pi ∈ N for all i = 1, . . . , I. Then there exist an absolute constant K p1 ... p I such that K p1 ... p I , (6) χ p1 p2 ... p I (Il11 l12 l13 , . . . , Il I 1 l I 2 l I 3 ) ≤ (2l11 + 1) p/4 where p := ( p1 + · · · + p I ) ≥ 4. As a consequence, for any k ∈ N, as l11 → ∞,
Il l l Il l l 11 12 13 , . . . , k1 k2 k3 lk1 lk2 lk3 l11 l12 l13
→d N (0, Ik ),
(7)
where Ik denotes the (k × k) identity matrix. Remark 2.1 The condition that (li1 , li2 , li3 ) = (li 1 , li 2 , li 3 ) whenever i = i is merely notational and entails no loss of generality; indeed, whenever (li1 , li2 , li3 ) = (li 1 , li 2 , li 3 ) it suffices to identify the two indexes and change the values of pi accordingly. Explicit expressions for K p1 ... p I are provided in Sect. 4. Proof The proof is lengthy and computationally burdensome, so before we proceed we find it useful to sketch heuristically its main features. The first step is to notice that, in view of the well-known diagram formula (see for instance [21]), the higher order cumulants of the bispectrum can be associated with sums over all possible connected graphs corresponding to cross-products of Wigner’s 3 j coefficients. The bound can then be obtained by partitioning these connected graphs into trees, that is subgraphs with no loops, and then associating these trees to the coefficients of some unitary matrices arising in matrix representation theory for S O(3). Let us now make this argument rigorous. We start by reviewing very rapidly some elementary notions from graph theory, which is widely used in physics when handling Wigner’s 3 j coefficients (see [22, Chap. 11]). Take i = 1, . . . , p and j = 1, . . . , qi , and consider the set of indexes: ⎧ ⎫ ⎨ (1, 1) . . . . . . (1, q1 ) ⎬ ... ... ... ... T = ; ⎩ ⎭ ( p, 1) . . . . . . ( p, q p ) we stress that the number of columns qi need not be the same for each row i. A diagram γ is any partition of the elements of T into pairs like {(i 1 , j1 ), (i 2 , j2 )} : these pairs are called the edges of the diagram. We label (T ) the family of these diagrams.
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We also note that if we identify each row i k with a vertex (or node), and view these vertexes as linked together by the edges {(i k , jk ), (i k , jk )} = i k i k , then it is possible to associate to each diagram a graph. As it is well known, a graph is an ordered pair (I, E) where I is non-empty (in our case the set of the rows of the diagram), and E is a set of unordered pairs of vertexes (in our cases, the pair of rows that are linked in a diagram). In the sequel, we follow a quite standard notation and we refer to Foulds [8], Surgailis [21] and Marinucci [18] for detailed definitions of basic concepts (such as flat edges, connected components, trees, k-loops). We need first to introduce a new set of triples L = {( 11 , 12 , 13 ) , . . . ( R1 , R2 , R3 )} , defined by ( r 1 , r 2 , r 3 ) : = (l11 , l12 , l13 ) for r = 1, . . . , p1 ( r 1 , r 2 , r 3 ) : = (l21 , l22 , l23 ) for r = p1 + 1, . . . , p1 + p2 ... ( r 1 , r 2 , r 3 ) : = (l I 1 , l I 2 , l I 3 ) for r = p1 + · · · + p I −1 + 1, . . . , p1 + · · · + p I ; more explicitly, the set L is obtained by replicating pi times each of the (li1 , li2 , li3 ) I pi ; triples. Let T be a set of indexes {(r, k)}, where k = 1, 2, 3 and r = 1, 2, . . . , i=1 recall that r k −m r k (8) Ea r1 k1 m r1 k1 a r2 k2 m r2 k2 = (−1)m r1 k1 C k1 δ r 2k 2 δm r1 k12 2 . 1 1
For any γ ∈ (T ), we can define
δ(γ ; L) := {
(ru ku ),(ru ku )
−m r k
}∈γ
r k
(−1)m ru ku δm ru kuu u δ r u k u ;
(9)
u u
for brevity’s sake, we write δ(γ ) rather than δ(γ ; L) causes no ambi whenever this guity. Write {(r, k), .} ∈ γ to signify that the pair (r, k), (r , k ) belongs to γ , for some (r , k ); for any diagram γ , we can hence define D(γ ) :=
r k
m r k =− r k r :(r,k)∈T {(r,k),.}∈γ
r 2 mr 2
r 1 mr 1
r 3 mr 3
δ(γ ).
(10)
The meaning of (9)–(10) is the following: in view of the orthogonality properties of the spherical harmonic coefficients, we must only focus on cross-products of Wigner’s coefficients such that the indexes (lr k , m r k ) match in pairs; if this requirement is not met, the corresponding summand is identically zero. We define also D(A) :=
γ ∈A
D(γ ) =
r k
γ ∈A m r k =− r k r :(r,k)∈T {(r,k),.}∈γ
1 mr 1
2 mr 2
3 mr 3
δ(γ );
in words, D(.) represents the component of the expected value that corresponds to the diagrams belonging to A.
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In view of the diagram formula and the multilinearity of cumulants, we obtain χ p1 p2 ... p I (Il11 l12 l13 , . . . , Il I 1 l I 2 l I 3 ) r k r 1 = mr 1 γ ∈C (T ) m r k =− r k r :(r,k)∈T {(r,k),.}∈γ
r 2 mr 2
r 3 mr 3
δ(γ ; L)
= D[C (T ); L], where by C (T ) we denote the (finite) set of connected diagrams over T. It is shown in Lemmas 3.1–3.3 in Marinucci [18] that diagrams with a 1-loop correspond to summands identically equal to zero, whereas diagrams with p nodes and loops of orders 2 or 3 can be reduced to terms corresponding to diagrams with p − 2 nodes times a factor O((2l11 + 1)−1 ). In the sequel, it is hence sufficient to focus only on diagrams which have no loops of orders 1, 2 or 3. Now call R the set of rows of T , and partition it into subsets such that R = R1 ∪ R2 ∪ · · · ∪ R g . Then it will also possible to partition γ into subdiagrams γ1 , γ2 , . . . , γg , γ12 , . . . , γg−1,g such that γ1 includes the pairs with both row indexes in R1 , γ2 includes the pairs with both rows in R2 , γ12 includes the pairs with one row in R1 and the other in R2 , and so on; we assume all internal subdiagrams γi to be non-empty, whereas this need not be the case for γi j . In terms of edges, γ1 includes the edges that are internal to R1 , γ2 includes the edges that are internal to R2 , γ12 includes the edges that connect R1 to R2 , and so forth. Note that, γ = (γ1 ∪ γ2 ∪ · · · ∪ γg ∪ γ12 ∪ · · · ∪ γg−1,g ) and we can write
D(γ ) =
g
g
r k
i=1 j=i+1 m r k =− r k {(r,k),.}∈γi j
× δ(γi j ) =
g g
⎧ ⎡ ⎪ ⎪ g r k ⎨ r 1 ⎢ ⎢ ⎣ mr 1 ⎪ ⎪ ⎩i=1 m r k =− r k r ∈Ri
r 3 mr 3
{(r,k),.}∈γi
r k
r 2 mr 2
i=1 j=i+1 m r k=− r k {(r,k),.}∈γi j
g
⎤⎫ ⎪ ⎪ ⎥⎬ δ(γi )⎥ ⎦⎪ ⎪ ⎭
X Ri ;γi
δ(γi j ),
i=1
where X Ri ;γi :=
r k
r 1 mr 1
m r k =− r k r ∈Ri {(r,k),.}∈γi
r 2 mr 2
r 3 mr 3
δ(γi ).
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Fig. 1 The partition of a graph into two subgraphs
X Ri ;γi can be viewed as a vector whose elements are indexed by m ri ,ki , where ri ∈ Ri and {(ri , ki ), .} ∈ / γi (indeed those indexes m ri ki such that {(ri , ki ), .} ∈ γi have been summed up internally). For instance, for g = 2 we have ⎧ ⎪ ⎪ ⎨
r k
D(γ ) =
m r k =− r k {(r,k),.}∈γ12
×
⎧ ⎪ ⎪ ⎨
r k
⎪ ⎪ ⎩ m r k =− r k
{(r,k),.}∈γ1
r k
⎪ ⎪ ⎩ m r k =− r k
{(r,k),.}∈γ2
r 1 mr 1
r 2 mr 2
r ∈R1
r 1 mr 1
r ∈R2
r 2 mr 2
r 3 mr 3
r 3 mr 3
δ(γ1 )
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
⎫ ⎪ ⎪ ⎬
δ(γ2 ) δ(γ12 ). ⎪ ⎪ ⎭
In Fig. 1, we provide a graph with eight nodes #(R) = 8 (right), and then (left) we partition it with g = 2, #(R1 ) = #(R2 ) = 4; the nodes in R1 are labelled with a circle, the nodes in R2 are labelled with a cross, the edges in γ1 and γ2 have a solid line while those in γ12 are dashed. Here we have 3 + 3 = 6 internal sums and six external ones. Assume now that γi does not include any loop, for i = 1, . . . , g. We shall show that g X R ;γ |D(γ )| ≤ (11) i i i=1
where . denotes Euclidean norm, and X R ;γ ≤ i i
(2 r k + 1)−1/2 ≤ (2
{(r,k),.}∈γi
min
{(r,k),.}∈γi
r k + 1)(#(Ri )−1)/2 ;
(12)
note that if γi does not include any loop the number of edges it contains must be identically equal to #(Ri ) − 1, where #(.) denotes the cardinality of a set. Let us consider (11) first. It is clear that we can choose new indexes such that X Ri ;γi =: X (i) is a vector with elements ! (i) X (i) = X m , − ≤ m ≤ , j = 1, . . . v ij ij ij i , i = 1, . . . , g; i1 ,...,m iv i
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" for this new set of indexes, namely we write T ⎧ ⎫ ⎨ (1, 1) . . . . . . . (1, v1 ) ⎬ "= ... ... ... ... T ; ⎩ ⎭ (g, 1) . . . . . . (g, vg )
here g can be viewed as the number of trees and vi as the number of vertexes which (1) (g) (i) are #vi in a given tree i. Clearly the vectors X , . . . , X have dimensions #(X ) = j=1 (2 i j + 1). The following Lemma can be viewed as a multivariate extension of the Cauchy–Schwartz inequality. " with Lemma 2.1 (Multivariate Cauchy–Schwartz inequality) Let " γ be a partition of T no flat edges. We have ⎧ g vi ⎨ ⎩
⎫ i j g $ g $ ⎬ $ (i) $ (i) γ )| ≤ $X m i1 ...m i vi $ |δ(" X ⎭
i=1 j=1 m i j =− i j
i=1
i=1
where ⎧ g vi ⎨ ⎩
⎫ i j ⎬
i=1 j=1 m i j =− i j
⎭
=
11 m 11 =− 11
gvg
...
.
m gvg =− gvg
Proof The result follows from the iterated application of the Cauchy–Schwartz inequality; we shall argue by induction. Without loss of generality, we can assume that the diagram γ is connected (if it is not, argue separately for the connected components). It is trivial to show that the result holds for g = 2, indeed in that case it just the standard Cauchy–Schwartz result. Let us now show that if the result holds for the product of g − 1 ≥ 2 components, it must hold for g components as well. Recall we consider diagrams with no flat edges, so the indexes cannot match on the same vector X (i) . Relabel terms so that there exist (at least) a link (that is, a common index m) between the first two vectors X (1) , X (2) . Without loss of generality we can order terms in such a way that the matching is internal for the first v ∗ indexes and external (that is, with the remaining nodes (3, 4, . . . , g)) for m 1 j : j = v ∗ + 1, . . . , v1 and m 2 j : γ =" γ12 ∪ " γ12 , where " γ12 is the set of edges linking j = v ∗ + 1, . . . , v2 . We write also " γ \" γ12 . We have node 1 to node 2 and " γ12 = "
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⎧ g vi ⎨ ⎩
⎫ i j g $ $ ⎬ $ (i) $ γ )| $X m i1 ...m i vi $ |δ(" ⎭
i=1 j=1 m i j =− i j
≤
⎧ ⎨ ⎩
i=1
1v ∗
2v2
···
m 1,v ∗ +1 =− 1v ∗
m 2v2 =− 2v2
⎫ ⎡⎡⎧ ⎬ ⎨ ⎣⎣ ⎭ ⎩
11
···
m 11 =− 11
m 1v ∗ =− 1v ∗
⎤ $ !$ $ $ (1) γ12 )| ⎦ X (2) × $ Xm $ |δ(" 11 ...m 1 v1 m 11 ...m 2 v2 ⎧ g vi ⎨
×
≤
⎩
⎭
i=3
⎫ ⎧⎡⎧ ⎫ ⎤1/2 1v ∗ 11 ⎬⎪ ⎨⎨ ⎬ !2 ⎦ ⎣ X (1) ··· ⎭⎪ ⎩ ⎭ m 11 ...m 1 v1 ⎩ =− 2v m 11 =− 11 m 1v ∗ =− 1v ∗
1v ∗
2v2
···
⎩
m 1,v ∗ +1 =− 1v ∗
⎡⎧ ⎨ ×⎣ ⎩
⎫ ⎬
⎫ ⎤ i j g $ $ $ ⎬ $ $ (i) $ γ12 )$⎦ $X m i1 ...m i vi $ $δ(" ⎭
i=3 j=1 m i j =− i j
⎧ ⎨
1v ∗
21
m 2v2
m 21 =− 21
⎫ ⎬
2v ∗
···
⎡⎧ g vi ⎨ ×⎣ ⎩
2
m 2v ∗ =− 2v ∗
⎭
(2) Xm 21 ...m 2 v2
!2
⎤1/2 ⎦
⎫ ⎫ ⎤ i j g $ $ ⎬ ⎬ $ $ $ (i) $ γ12 )$ . $ X m i1 ...m i vi $ ⎦ $δ(" ⎭ ⎭
i=3 j=1 m i j =− i j
(13)
i=3
The last step follows again by standard Cauchy–Schwartz inequality. Now define
(1;2) Xm 1,v ∗ +1 ...m 2 v2
⎡⎧ ⎨ := ⎣ ⎩
11
···
m 11 =− 11
⎡⎧ ⎨ ×⎣ ⎩
1v ∗ m 1v ∗ =− 1v ∗
21
(1) Xm 11 ...m 1 v1
⎭
⎫ ⎬
2v ∗
···
m 21 =− 21
⎫ ⎬
m 2v ∗ =− 2v ∗
⎭
!2
(2) Xm 21 ...m 2 v2
⎤1/2 ⎦ !2
⎤1/2 ⎦
so that (13) becomes
1v ∗
···
m 1,v ∗ +1 =− 1v ∗
×
31
m 2v2 =− 2v2 m 31 =− 31
qvg
···
m gvg =− gvg
(1;2) Xm 1,v ∗ +1 ...m 2 v2
g $ $ $ (1;2) $ (i) $ (i) γ12 )$ ≤ X m $X m i1 ...m i vi $ $δ(" X 1,v ∗ +1 ...m 2 v2
g $ i=3
123
2v2
i=3
(14)
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by the inductive step (indeed within the curly brackets we have the product of g − 2 + 1 = g − 1 components). Now notice that ⎧ ⎫1/2 2v2 1v ∗ ⎨ % &2 ⎬ (1;2) (1;2) Xm ··· X m 1,v∗ +1 ...m 2 v2 = 1,v ∗ +1 ...m 2 v2 ⎩ ⎭ m 1,v ∗ +1 =− 1v ∗ m 2v2 =− 2v2 ⎧ ⎧ ⎫ 2v2 1v ∗ 1v ∗ 11 ⎨ ⎨ ⎬ = ··· ··· ⎩ ⎩ ⎭ m 1,v ∗ +1 =− 1v ∗
(1) × Xm 11 ...m 1 v1
=
⎧ ⎨ ⎩ ×
21
···
m 21 =− 21
1v1
···
m 11 =− 11
⎩
⎩
m 11 =− 11
21 m 21 =− 21
m 1v1 =− 1v1
m 2v ∗ =− 2v ∗
(1) Xm 11 ...m 1 v1
···
m 2v2 =− 2v2
⎭
(2) Xm 21 ...m 2 v2
⎫1/2 !2 ⎬ ⎭
⎫1/2 !2 ⎬
2v2
m 1v ∗ =− 1v ∗
⎫ ⎬
2v ∗
11
⎧ ⎨
m 2v2 =− 2v2
⎧ !2 ⎨
(2) Xm 21 ...m 2 v2
⎭ ⎫1/2 !2 ⎬ ⎭
= X (1) × X (2) ,
and thus by substitution into (14) the proof is completed.
It is not difficult to see that (11) is an immediate consequence of Lemma 2.1; in particular, note that " γ can be viewed as the diagram which is obtained from γ by identifying all nodes that belong to the same set Ri , for i = 1, . . . , g. Now let us consider (12). The following result uses the previous inequality to bound the components of a connected graph. Lemma 2.2 (a) Every connected graph with no loops of orders 1,2 or 3 can be partitioned as γ = (γ1 ∪ γ2 ∪ · · · ∪ γg ∪ γ12 ∪ · · · ∪ γg−1,g ) where γi has no loops of any order and is such that #(Ri ) ≥ 2 (in other words, γ can be broken into into binary trees with at least two nodes) (a) For all γi , i = 1, . . . , g we have X R ;γ ≤ i i
{(r,k),.}∈γi
(2 r k + 1)
−1/2
≤ 2
min
{(r,k),.}∈γi
r k
(#(Ri )−1)/2 +1
that is, every tree with p nodes corresponds to summands which are O((2 11 + 1)−( p−1)/2 ), where p is the number of nodes in the trees.
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(a) For all connected graphs γ with p nodes, we have
|D[γ ; L]| ≤
p/4
(2 r 1 + 1)−1 .
(15)
r =1
Proof (a) We cut edges till we reach the point where there are only binary trees or isolated points. Each of these isolated points can be connected to either another isolated point, in which case we simply have a tree with two nodes, or to a binary tree. The graph which is obtained by linking this point to the tree is itself a tree if its paths have at most two edges: recall there are no loops of order 2 or 3. On the other hand, if the graph has a path which covers four nodes, then we cut two edges and obtain two trees with (at least) two nodes. These procedures can be iterated until no isolated point remains. (b) We recall the identity 1 m1
m 1 ,m 2
2 m2
λ1 µ1
2 m2
1 m1
λ1 µ1
λ µ
=
δλ11 δµ11 2λ1 + 1
.
It follows that any cross-product of p Wigner’s 3 j coefficients corresponding to a binary tree can be bounded by a term like $ $ λ p−1 $ λ1 2 λ1 λ2 1 λ1 3 $ · · · $ m 1 m 2 µ1 µ1 m 3 µ2 µ p−1 =−λ p−1 m 1 ,...m p ;m $µ1 =−λ1 $ $2 $ λ p−1 p+1 $ ··· µ p−1 m p+1 m $$ ⎧ λ1 λ1 ⎨ 1 ⎪ 2 λ1 2 λ1 1 = m 1 m 2 µ1 m 1 m 2 µ1 ⎪ m 3 ,...m p ;m ⎩ µ1 =−λ1 µ1 =−λ1 m 1 ,m 2 ⎫ ⎪ λ p−1 λ p−1 λ p−1 p+1 ⎬ λ ··· · · · p−1 p+1 µp−1 m p+1 m ⎪ µ p−1 m p+1 m ⎭
µ p−1 =−λ p−1 µp−1 =−λ p−1
=
⎪
λ1
m 3 ,...m p ;m ⎩µ1 ,µ1 =−λ1
···
123
⎧ ⎪ ⎨
λ p−1 µ p−1
µ
δµ11 ··· 2λ1 + 1
p+1 m p+1
m
λ p−1
λ p−1
µ p−1 =−λ p−1 µp−1 =−λ p−1
λ p−1 µp−1
p+1 m p+1
⎫ ⎪ ⎬ m ⎪ ⎭
A central limit theorem and higher order results for the angular bispectrum
401
and iterating $ $ λ1 λ p−1 $ 2 λ1 1 λ1 $ ... $ m m µ µ 1 2 1 1 µ p−1 =−λ p−1 m 1 ,···m p ;m $µ1 =−λ1 ⎫−1 $2 ⎧ $ ⎨ p−1 ⎬ $ λ p−1 p+1 $ = (2λ + 1) . ··· j µ p−1 m p+1 m $$ ⎩ ⎭
3 m3
λ2 µ2
(16)
j=1
(c) Connected components with loops of order 1, 2 or 3 can be reduced as shown in Marinucci [18]. From part (b), we have |D[γ ; L]| ≤
( p−g)/2
(2 r 1 + 1)−1 ,
(17)
r =1
whereas from (a) we learn that it is always possible to choose a partition such that g ≤ p/2; the result follows immediately.
Remark 2.2 The identity (16) can be given a nice interpretation in terms of matrix representation theory. Define ⎫ ⎧ λ p−1 λ1 ⎬ ⎨p−1 2 λ1 λ2 1 λ1 3 (2λ j + 1) ··· m 1 m 2 µ1 µ1 m 3 µ2 ⎭ ⎩ µ1 =−λ1 µ p−1 =−λ p−1 j=1 λ p−1 p+1 λ ,λ ,...,λ ;µ ; =: C 11,m21 ;...; p−1 ··· pm p µ p−1 m p+1 m λ ,λ ,...,λ
;µ
(which could be labelled multiit can be shown that the coefficients C 11,m21 ;...; p−1 pm p variate Clebsch–Gordan, see [23], [22, Chap. 8]) are the elements of unitary matrices (=: C 1 ... p , say) intertwining multiple direct sums and tensor product (reducible) representations of the group of rotations S O(3). The relationships m 1 ,...m p
λ ,λ ,...,λ
! ;µ 2
C 11,m21 ;...; p−1 pm p
=
λ1
···
λ p−1
λ ,λ ,...,λ
! ;µ 2
C 11,m21 ;...; p−1 pm p
λ p−2 λ p−1 µ=−λ p−1
=1
are then an immediate consequence of the unitary property. Remark 2.3 An inspection of our argument [especially in part (c)] reveals that the bound (15) [and hence in (6)] can be improved in those cases where it is possible to partition the graph γ into a minimal number of trees. For instance, it is known (see for instance [4]) that all connected graphs with up to 8 nodes can be partitioned into two binary trees; for such cases, we hence obtain the bound χ p1 ... p I (Il11 l12 l13 , . . . , Il I 1 l I 2 l I 3 ) ≤
K p1 ... p I p
(2 11 + 1) 2 −1
,
p ≤ 8.
(18)
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For p ≥ 10 it is no longer true that such a partition necessarily exists; we leave for future research, however, to ascertain whether (6) can be improved by a more efficient use of graphical arguments. 3 Unknown angular power spectrum We consider now the case where the angular power spectrum is unknown and estimated from the data. Here, the main technical problem is the following: given the random normalization in (5), the bispectrum is no longer a linear combination of products of Gaussian variates, and hence we are unable to use a neat expression for its cumulants as in the previous section. In this section we shall hence focus on moments, rather than cumulants: this is merely a matter of convenience, given the one-to-one relationship which exists between these two set of quantities. Recall we are assuming as before (li1 , li2 , li3 ) = (li 1 , li 2 , li 3 ) whenever i = i ; as a simple consequence of (6) and a standard expansion of higher order moments into cumulants [21, Eq. (2.2)] we obtain the uniform bound $ I $ pi I $ $ K p1 ... p I Ili1 li2 li3 $ $ {( pi − 1)!!}$ ≤ , some K p1 ... p I > 0, − $E $ $ 2l11 + 1 l l l i1 i2 i3 i=1 i=1 (19) where ' ( p − 1)!! :=
( p − 1) × ( p − 3) × · · · × 1, for p even . 0, for p odd
Our aim will be to establish an analogous expression under random normalization, and then exploit it to extends Theorem 2.1 to the case where the angular power spectrum is estimated from the data. Theorem 3.1 Assume that (li1 , li2 , li3 ) = (li 1 , li 2 , li 3 ) whenever i = i , where I ≥ 1 and pi ∈ N for all i = 1, . . . , I. There exist an absolute constant K p1 ... p I > 0 such that $ I $ pi I $ $ K p1 ... p I Ili1 li2 li3 $ $ {( pi − 1)!!}$ ≤ . (20) − $E $ $ 2l11 + 1 li1 li2 li3 i=1 i=1 As a consequence, for any k ∈ N, as l11 → ∞,
Il l l Il l l 11 12 13 , . . . , k1 k2 k3 ll1 lk2 lk3 l11 l12 l13
→d N (0, Ik ),
(21)
where Ik denotes the (k × k) identity matrix. Proof We need first to establish some more notation. Partition the indexes { 11 , 12 , 13 , . . ., R1 , R2 , R3 } into equivalence classes where r k takes the same value; we
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403
can then relabel them as u , u = 1, . . . , c where c is the number of such classes, i.e., the number of different values of ; clearly c ≤ 3R. For convenience, we write [{(l11 , l12 , l13 ), . . . (l I 1 , l I 2 , l I 3 ); p1 , . . . , p I }] := { 11 , 12 , 13 , . . . , R1 , R2 , R3 } , and define G [{(l11 , l12 , l13 ), . . . (l I 1 , l I 2 , l I 3 ); p1 , . . . , p I }] := =
c #(u) g u ; 2 u=1 c (#(u))/2 2 u + 1 u=1
j=1
2 u + 2 j − 1
,
where #(u) denotes the cardinality of the class where u belongs, or in other words the number of times that each particular value is repeated in { 11 , 12 , 13 , . . . , R1 , R2 , R3 }. We give some example to make the previous definition more transparent. Consider for notational simplicity the univariate case I = 1. For l1 < l2 < l3 we have c = 3 and G [{(l11 , l12 , l13 ), . . . (l I 1 , l I 2 , l I 3 ); p1 , . . . , p I }] =
R ' 3 u=1 j=1
( 2lu + 1 ; 2lu + 2 j − 1
for l1 = l2 < l3 we have c = 2 and G [{(l11 , l12 , l13 ), . . . (l I 1 , l I 2 , l I 3 ); p1 , . . . , p I }] ( ( 2R ' R ' 2l1 + 1 2l3 + 1 ; = 2l1 + 2 j − 1 2l3 + 2 j − 1 j=1
j=1
for l1 = l2 = l3 we have c = 1 and G [{(l11 , l12 , l13 ), . . . (l I 1 , l I 2 , l I 3 ); p1 , . . . , p I }] =
3R ' j=1
( 2l1 + 1 . 2l1 + 2 j − 1
Take p1 , . . . , p I ∈ N: by exactly the same argument as in Marinucci [18, Theorem 4.1], it is immediate to obtain E
I i=1
p Ili1ili2 li3
=E
I
pi
Ili1 li2 li3 G [{(l11 , l12 , l13 ), . . .(l I 1 , l I 2 , l I 3 ); p1 , . . . , p I }] .
i=1
(22)
123
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D. Marinucci
Notice that for p (= p1 = · · · + p I ) odd the equality holds trivially as both sides are zero by symmetry arguments; also lim G [{(l11 , l12 , l13 ), . . . (l I 1 , l I 2 , l I 3 ); p1 , . . . , p I }] = 1.
l11 →∞
It is now sufficient to combine Theorem 3.1 and (22) to show that $ I pi I $ Ili1 li2 li3 $ − {( pi − 1)!!}G[{(l11 , l12 , l13 ), $E $ li1 li2 li3 i=1 i=1 $ $ K p1 ... p I $ . . . .(l I 1 , l I 2 , l I 3 ); p1 ,. . . , p I }]$ ≤ $ 2l11 + 1 It is immediate to see that, for some K p1 ... p I > 0 |G [{(l11 , l12 , l13 ), . . . (l I 1 , l I 2 , l I 3 ); p1 , . . . , p I }] − 1| ≤
K p1 ... p I 2l11 + 1
whence (20) follows. The CLT is then an immediate consequence of the methods of moments.
Remark 3.1 Results (19)–(20) provide the rate of convergence of the moments of the bispectrum to the Gaussian values under random and non-random normalizations. This settles some questions raised in Komatsu et al. [12], where the moments of Ili1 li2 li3 and Ili1 li2 li3 where compared by means of Monte Carlo simulations. Remark 3.2 It is interesting to note that the angular bispectrum ordinates at different multipoles are asymptotically independent, for any triples (l11 , l12 , l13 ) = (l21 , l22 , l23 ). Theorems 2.1–3.1 show that a (multivariate) Central Limit Theorem holds for the angular bispectrum in the high resolution sense, when all the multipoles (l1 , l2 , l3 ) considered diverge to infinity. We note that this case is indeed relevant for many cosmological applications, for instance for all equilateral models of non-Gaussianity [3]. It is also to be stressed how simulations have shown that a Central Limit Theorem can provide a useful approximation even for l ’s in the order of 10 [12], whereas available data reach l 103 . The next section is devoted to higher order results, by which we can provide quite accurate approximations of the bispectrum moments even for very small values of the multipoles. 4 Higher order results For notational simplicity, in this section we shall focus on the univariate case, that is, we do not consider cross-moments; the multivariate generalization is straightforward,
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A central limit theorem and higher order results for the angular bispectrum
405
however, and no new ideas are required. In the statement of the Theorem to follow, we use the Wigner’s 6 j symbols, which are defined by ' ( a b e c d f a b e c d e a d f c b f , (−1)e+ f +ε+φ := α β ε γ δ −ε α δ −φ γ β φ α,β,γ ε,δ,φ
(23) see [22, Chap. 9] for a full set of properties; we simply recall here that $' ($ $ abe $ $ $ $ c d f $ 1 1 1 ≤ min √ . ,√ ,√ (2e + 1)(2 f + 1) (2a + 1)(2c + 1) (2b + 1)(2d + 1) (24) Theorem 4.1 There exist an absolute constant K p > 0 such that, for all integer p ≥ 4 $ $ p $ $ Il1 l2 l3 p( p − 2) Il1 l2 l3 $ $ − ( p − 1)!! − ( p − 1)!!$ χ4 $E $ $ 24 l1 l2 l3 l1 l2 l3 ≤ where
Kp , (2l1 + 1)2
Il1 l2 l3 χ4 := li1 li2 li3 Il1 l1 l2 χ4 := l1 l1 l2 Illl := χ4 √ lll
6 6 6 + + +6 2l1 + 1 2l2 + 1 2l3 + 1 ' 24 96 l1 + + 48 l 2l1 + 1 2l2 + 1 1 ' ( 2 6 × 18 l l l . + 64 l l l 2l + 1
(25)
( l1 l2 l3 , for l1 < l2 < l3 , l1 l2 l3 ( l1 l2 , for l1 = l2 . l1 l2
'
Proof As discussed earlier, for p odd the right-hand side of (25) is identically zero. For p even, we express once again higher order moments by means of cumulants (see again [21, Sect. 2]); a direct combinatorial argument yields the following expression
p Il1 l2 l3 Il1 l1 l2 1 p p p−4 E = ( p − 1)!! + ( p − 5)!! + χ4 4 4 2 4 l1 l2 l3 l1 l1 l2 I I I p l l l l l l l l l ( p−9)!!+ ( p−7)!! + · · ·+χ p 1 1 2 . χ6 1 1 2 × χ42 1 1 2 6 l1 l1 l2 l1 l1 l2 l1 l1 l2
123
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D. Marinucci
Fig. 2 Loops of order two (left) and three (right)
I √ l1 l1 l2 l1 l1 l2
= O((2l1 + 1)−2 ) for I u ≥ 6, whence we need only provide an explicit evaluation for χ4 √l1 l1 l2 . For
Now from Theorem 2.1 and (18) we know that χu
l1 l1 l2
the latter, note that connected graphs with four nodes must include a loop of order 1, 2 or 3. Terms with a 1-loop are identically zero [18, Lemma 3.1]. Graphs associated to terms with 2- or 3-loops are represented in Fig. 2. We consider the three cases separately. (a) For l1 < l2 < l3 , we consider first the graphs with loops of order two (on the left-hand side of the Fig. 2). By means of Lemma 3.2 in Marinucci [18], it is simple to ascertain that each corresponding term produces a factor (2li + 1), where the li correspond to the index which is not in the two-loop (the dashed line). Direct counting of permutations shows that there are 6 such graphs for each fixed li . Likewise, the Wigner’s 6 j coefficients arise in connection with the graphs with loops of order three where each node is linked to all three others, compare (23) and Lemma 3.3 in Marinucci [18]; direct counting of possible permutations shows that there can be six distinct combinations of this form (fix for instance node 1, which by assumption is linked to all the other three nodes: there are three degrees of freedom to choose the connection with node 2, then two left for node 3, for a total of six, as claimed). (b) For l1 = l2 = l3 , there are two possible types of graphs with loops of order two, that is, those where both 2-loops involve l2 and those where one 2-loop involves l2 and the other l3 . In the former case, there are three ways to choose the pairs, two ways in each pair to choose the links, and two ways to choose the way to match the edges corresponding to l3 : the total is 24. In the latter case, it can be checked that there are 6 possible choices of pairs and 4 possible matchings for the 2-loop which involves l2 and l3 . The remaining term is similar. (c) For l1 = l2 = l3 , again we choose in three possible ways the pairs, plus the single link in two possible ways; within each pair, we can choose 3 × 3 couples and two possible ways to link them. The remaining term is similar.
Remark 4.1 Theorem 4.1 can be used to establish (in a merely formal sense) Edgeworth approximations for the asymptotic behavior of the angular bispectrum (see [9]). We provide here only some heuristic discussion, and leave for future research the possibility to establish rigorously a valid Edgeworth expansion. Let (2 p) µl1 l2 l3
123
:= E
2p
Il1 l2 l3 p
l1 l2 l3
,
A central limit theorem and higher order results for the angular bispectrum
407
whence ∞
∞ 1 (it)2 p (2 p − 1)!! + χ4 2 p! 2 p! 24
(2 p) (it) µl1 l2 l3
p=1
2p
p=0
×2 p(2 p − 2)(2 p − 1)!! +
Il l l 112 l1 l1 l2
O(l1−2 )
∞ (it)2 p 1 = (2 p − 1)!! + χ4 2 p! 24
p=0 ∞
Il l l 112 l1 l1 l2
∞ (it)2 p 2 p! p=2
(it)4
(it)2 p (2 p − 1)!! + O(l1−2 ) 2 p! p=0 2 Il1 l1 l2 −t 1 = exp t 4 + O(l1−2 ). 1 + χ4 2 24 l1 l1 l2 ×
Again in a formal sense, we can take Fourier transforms on both sides, leading to the conjecture that Pr
x x Il1 l2 l3 1 Il1 l1 l2 ≤ x = φ(z)dz + χ4 (z 4 −6z +3)φ(z)dz + O(l1−2 ), 24 l1 l2 l3 l1 l1 l2 −∞
−∞
where φ(z) denotes the density function of a standard Gaussian variable. Remark 4.2 It is interesting to note that for p = 2 (25) holds exactly, i.e., with K p ≡ 0. We provide hence an explicit evaluation of these moments. First recall that [22, Chap. 9] '
l1 l1 ×
l2 l2
l3 l3
2l 2 +2l3 n=l1 +l2 +l3
)
( =
{(l1 + l2 − l3 )!(l1 + l3 − l2 )!(l2 + l3 − l1 )!}3 (l1 + l2 + l3 + 1)!
*2
(−1)n (n + 1)! . ((n − l1 − l2 − l3 )!)4 ((2l1 + 2l2 − n)!)((2l1 + 2l3 − n)!)((2l2 + 2l3 − n)!)
This expression can be simplified for some values of the triple (l1 , l2 , l3 ). More precisely, we have '
l l
l l
l l
( =
4l n=3l
(−1)n (n + 1)! ((n − 3l)!)4 ((4l − n)!)3
and [22, Eq. 9.5.2.10] '
l1 l1
l2 l2
l1 + l2 l1 + l2
( =
(2l1 )!(2l2 )! . (2l1 + 2l2 + 1)!
123
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D. Marinucci
Thus we have 6 6 6 (2l1 )!(2l2 )! + + +6 , l1 = l2 , 2l1 + 1 2l2 + 1 2l1 + 2l2 + 1 (2l1 + 2l2 + 1)! (26) 2 24 ((2l)!) 96 + + 48 , (27) = 3 × 22 + 2l + 1 4l + 1 (4l + 1)!
EIl41 l2 ,l1 +l2 = 3+
4 EIll,2l
and 4 EIlll
4l (l!)6 (−1)n (n + 1)! 6 × 182 4 =3×6 + . +6 2l + 1 ((3l + 1)!)2 ((n − 3l)!)4 ((4l − n)!)3 2
(28)
n=3l
The validity of (26)–(28) has been confirmed with a remarkable accuracy by a Monte Carlo experiment (not reported here), where 200 replications of the sample bispectrum at various multipoles (l1 , l2 , l3 ) were generated for Gaussian fields; their sample moments were then evaluated and found to be in excellent agreement with our theoretical results. By means of (22) these results can be immediately extended to the case where the normalization is random. More precisely, we obtain for l1 = l2 ( ' 6 6 (2l1 )!(2l2 )! 6 + + +6 E Il41 l2 ,l1 +l2 = 3 + 2l1 + 1 2l2 + 1 2l1 + 2l2 + 1 (2l1 + 2l2 + 1)! ( ' 2l1 + 1 2l2 + 1 2l3 + 1 , × 2l1 + 3 2l2 + 3 2l3 + 3 (' ( ' 24 ((2l)!)2 2l + 1 2l + 1 2l + 1 4l + 1 96 4 + +48 , E Ill,2l = 12+ 2l + 1 4l + 1 (4l + 1)! 2l + 3 2l + 5 2l + 7 4l + 3 and )
4 E Illl
4l (l!)6 (−1)n (n + 1)! 6 × 182 + 64 = 3 × 36 + 2 2l + 1 ((3l + 1)!) ((n − 3l)!)4 ((4l − n)!)3
×
6 ' k=1
(
*
n=3l
2l + 1 . 2l + 2k − 1
These expressions can be of practical interest for statistical inference on cosmological data. For instance, the square bispectrum is often used in goodness-of-fit statistics to test the validity of the Gaussian assumption. The previous results yield immediately its exact variance, which so far has been typically evaluated by Monte Carlo simulations (see for instance [12]).
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