MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX

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1 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX J. WILLIAM HELTON, JEAN B. LASSERRE, AND MIHAI PUTINAR Abstract. We investigate an iscuss when the inverse of a multivariate truncate moment matrix of a measure µ has zeros in some prescribe entries. We escribe precisely which pattern of these zeroes correspons to inepenence, namely, the measure having a prouct structure. A more refine fining is that the key factor forcing a zero entry in this inverse matrix is a certain conitional triangularity property of the orthogonal polynomials associate with the measure µ. 1. Introuction It is well known that zeros in off-iagonal entries of the inverse M 1 of a n n covariance matrix M ientify pairs of ranom variables that have no partial correlation (an so are conitionally inepenent in case of normally istribute vectors); see e.g. Wittaker[7, Cor ]. Allowing zeros in the off-iagonal entries of M 1 is particularly useful for Bayesian estimation of regression moels in statistics, an is calle Bayesian covariance selection. Inee, estimating a covariance matrix is a ifficult problem for large number of variables, an exploiting sparsity in M 1 may yiel efficient methos for Graphical Gaussian Moels (GGM). For more etails, the intereste reaer is referre to Cripps et al. [3], an the many references therein. The covariance matrix can be thought of as a matrix whose entries are secon moments of a measure. This paper focuses on the truncate moment matrices, M, consisting of moments up to an orer etermine by. First we escribe precisely the pattern of zeroes of M 1 resulting from the measure having a prouct type structure. Next we turn to the stuy of a particular entry of M 1 being zero. We fin that the key is what we call the conitional triangularity property of orthonormal polynomials (OP) up to egree 2, associate with the measure. To give the flavor of what 1991 Mathematics Subject Classification. 52A20 52A. Key wors an phrases. Moment matrix; orthogonal polynomials. This work was initiate at Banff Research Station (Canaa) uring a workshop on Positive Polynomials, an complete at the Institute for Mathematics an its Applications (IMA, Minneapolis). All three authors wish to thank both institutes for their excellent working conitions an financial support. J.W. Helton was partially supporte by the NSF an For Motor Co., M. Putinar by the NSF an J.B. Lasserre by the (French) ANR. 1

2 2 HELTON, LASSERRE, AND PUTINAR this means, let for instance µ be the joint istribution µ of n ranom variables X = (X 1,..., X n ), an let {p σ } R[X] be its associate family of orthonormal polynomials. When (X k ) k i,j is fixe, they can be viewe as polynomials in R[X i, X j ]. If in oing so they exhibit a triangular structure (whence the name conitional triangularity w.r.t. (X k ) k i,j ), then entries of M 1 at precise locations vanish. Conversely, if these precise entries of M 1 vanish (robustly to perturbation), then the conitional triangularity w.r.t. (X k ) k i,j hols. An so, for the covariance matrix case = 1, this conitional triangularity property is equivalent to the zero partial correlation property well stuie in statistics (whereas in general, conitional inepenence is not etecte by zeros in the inverse of the covariance matrix). Interestingly, in a ifferent irection, one may relate this issue with a constraine matrix completion problem. Namely, given that the entries of M corresponing to marginals of the linear functional w.r.t. one variable at a time, are fixe, complete the missing entries with values that make M positive efinite. This is a constraine matrix completion problem as one has to respect the moment matrix structure when filling up the missing entries. Usually, for the classical matrix completion problem with no constraint on M, the solution which maxmizes an appropriate entropy, gives zeros to entries of M 1 corresponing to missing entries of M. But uner the aitional constraint of respecting the moment matrix structure, the maximum entropy solution oes not always fill in M 1 with zeros at the corresponing entries (as seen in examples by the authors). Therefore, any solution of this constraine matrix completion problem oes not always maximize the entropy. Its physical or probabilistic interpretation is still to be unerstoo. We point out another accomplishment of this paper. More generally than working with a measure is working with a linear functional l on the space of polynomials. One can consier moments with respect to l an moment matrices. Our results hol at this level of generality. 2. Notation an efinitions For a real symmetric matrix A R n n, the notation A 0 (resp. A 0) stans for A positive efinite (resp. semiefinite), an for a matrix B, let B or B T enote its transpose Monomials an polynomials. We now iscuss monomials at some length, since they are use in many ways, even to inex the moment matrices which are the subject of this paper. Let N enote the nonnegative integers an N n enote n tuples of them an for α = (α 1, α 2 α n ) N n, efine α := i α i. The set N n sits in one to one corresponence with the monomials via α N n X α := X α 1 1 Xα 2 2 X αn n.

3 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX 3 Recall also the stanar notation eg X α = α = α α n. By abuse of notation we will freely interchange below X α with α, for instance speaking about eg α rather than eg X α, an so on. Define the the fully grae partial orer (FG orer), on monomials, or equivalently for γ, α N n by γ α, iff γ j α j for all j = 1,, g. Important to us is: α β iff X α ivies X β. Define the the grae lexicographic orer (GLex) < gl, on monomials, or equivalently for γ, α N n first by using eg γ eg α to create a partial orer. Next refine this to a total orer by breaking ties in two monomials X m 1, X m 2 of same egree m 1 = m 2, as woul a ictionary with X 1 = a, X 2 = b,. Beware γ < gl α oes not imply γ α; for example, (1, 1, 3) < gl (1, 3, 1) but fails. However, β α an β α implies β < gl α. It is convenient to list all monomials as an infinite vector v (X) := (X α ) α N n where the entries are liste in GLex orer, henceforth calle the tautological vector; v (X) = (X α ) α R s() enotes the finite vector consisting of the part of v (X) containing exactly the egree monomials. Let R[X] enote the ring of real polynomials in the variables X 1,..., X n an let R [X] R[X] be the R-vector space of polynomials of egree at most. A polynomial is a linear combination of polynomials an g R[X] can be written (2.1) g(x) = g α X α = g, v (X) α N n for some real vector g = (g α ), where the latter is the stanar non-egenerate pairing between R s() an R s() R R[X] Moment matrix. Let y = (y α ) α N n (i.e. a sequence inexe with respect to the orering GLex), an efine the linear functional L y : R[X] R to be: (2.2) g L y (g) := α N n g α y α, whenever g is as in (2.1). Given a sequence y = (y α ) α N n, the moment matrix M (y) associate with y has its rows an columns inexe by α, α, an M (y)(α, β) := L y (X α X β ) = y α+β, α, β N n with α, β.

4 4 HELTON, LASSERRE, AND PUTINAR For example, M 2 (y) is 1 X 1 X 2 X 2 1 X 1 X 2 X 2 2 M 2 (y) : 1 1 y 10 y 01 y 20 y 11 y 02 X 1 y 10 y 20 y 11 y 30 y 21 y 12 X 2 y 01 y 11 y 02 y 21 y 12 y 03 X1 2 y 20 y 30 y 21 y 40 y 31 y 22 X 1 X 2 y 11 y 21 y 12 y 31 y 22 y 13 X2 2 y 02 y 12 y 03 y 22 y 13 y 04 A sequence y is sai to have a representing measure µ on R n if y α = x α µ(x), α N n, R n assuming of course that the signe measure µ ecays fast enough at infinity, so that all monomials are integrable with respect to its total variation. Note that the functional L y prouces for every 0 a positive semiefinite moment matrix M (y) if an only if L y (P 2 ) 0, P R[X]. The associate matrices M (y) are positive efinite, for all 0 if an only if L y (P 2 ) = 0 implies P = 0. We will istinguish this latter case by calling L y a positive functional. 3. Measures of prouct form In this section, given a sequence y = (y α ) inexe by α, α 2, we investigate some properties of the the inverse M (y) 1 of a positive efinite moment matrix M (y) when entries of the latter satisfy a prouct form property. Definition 1. We say that the variables X = (X 1,..., X n ) are mutually inepenent, an the moment matrix M (y) 0 has the prouct form property, if (3.1) L y (X α ) = n i=1 L y (X α i i ), α N n, α 2. If M (y) 1 (α, β) = 0 for every y such that M (y) 0 has the prouct form property, then we say the pair (α, β) is a congenital zero for -moments. We can now state an intermeiate result. Theorem 1. The pair α, β is a congenital zero for the - moment problem if an only if the least common multiple X η of X α an X β has egree bigger than.

5 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX 5 The above result can be conveniently rephrase in terms the max operation efine for α, β N n by max(α, β) := (max(α j, β j )) j=1,,n. Set η := max(α, β). Simple observations about this operation are: (1) X η is the least common multiple of X α an X β, (2) X α ivies X β iff X β = X η, X β ivies X α iff X α = X η, (3) η = n j=1 max(α j, β j ). Thus, Theorem 1 asserts that the entry (α, β) oes not correspon to a congenital zero in the matrix M (y) 1 if an only if max(α, β). Later in Theorem 4 we show that this LCM (least common multiple) characterization of zeros in M 1 is equivalent to a highly triangular structure of orthonormal polynomials associate with prouct measures. Example. In the case of M2 1 (y) in two variables X 1, X 2 we inicate below which entries M 2 (y) 1 (α, β) with β = 2, are congenital zeroes. These (α, β) inex the last three columns of M 2 (y) 1 an are X 2 1 X 1 X 2 X X 1 0 X 2 0 X X 1 X X Here means that the corresponing entry can be ifferent from zero. Note each correspons to X α failing to ivie X β. The proof relies on properties of orthogonal polynomials, so we begin by explaining in some etail the framework Orthonormal polynomials. A functional analytic viewpoint to polynomials is expeitious, so we begin with that. Let s() := ( ) n+ be the imension of vector space R [X]. Let, R s() enote the stanar inner prouct on R s(). Let f, h R[X] be the polynomials f(x) = s() α =0 f αx α an h(x) = s() α =0 h αx α. Then, f, h y := f, M (y)h R s() = L y (f(x)h(x)), efines a scalar prouct in R [X], provie M (y) is positive efinite. With a given y = (y α ) such that M (y) 0, one may associate a unique family (p α ) s() α =0 of orthonormal polynomials. That is, the p α s satisfy:

6 6 HELTON, LASSERRE, AND PUTINAR (3.2) p α lin.span{x β ; β gl α}, p α, p β y = δ αβ, α, β, p α, X β y = 0, if β < gl α, p α, X α y > 0, Note p α, X β y = 0, if β α an α β, since the latter implies β < gl α. Existence an uniqueness of such a family is guarantee by the Gram- Schmit orthonormalization process following the GLex orer on the monomials, an by the positivity of the moment (covariance) matrix, see for instance [1, Theorem , p. 68]. Computation. Suppose that we want to compute the orthonormal polynomials p σ for some inex σ. Then procee as follows: buil up the submoment matrix M (σ) (y) with columns inexe by all monomials β gl σ, an rows inexe by all monomials α < gl σ. Hence, M (σ) (y) has one row less than columns. Next, complete M (σ) (y) with an aitional last row escribe by [M (σ) (y)] σ,β = X β, for all β gl σ. Then up to a normalizing constant, p σ is nothing less than et (M σ (y)). To see this, let γ < gl σ. Then X γ p σ (X) µ(x) = et (B σ )(y), where the matrix B σ (y) is the same as M σ (y) except for the last row which is now the vector (L y (X γ+α )) α gl σ, alreay present in one of the rows above. Therefore, et (B σ )(y) = 0. Next, writing p σ (X) = ρ σβ X β, β gl σ its coefficient ρ σβ is just (again up to a normalizing constant) the cofactor of the element [M σ (y)] 1,β in the square matrix M σ (y) with rows an columns both inexe with α gl σ. Further properties. Now we give further properties of the orthonormal polynomials. Consier first one variable polynomials. The orthogonal polynomials p k have, by their very efinition a triangular form, namely p k (X 1 ) := l k ρ kl X l 1. The orthonormal polynomials inherit the prouct form property of M r (y), assuming that the latter hols. Namely, each orthonormal polynomial p α is a prouct (3.3) p α (X) = p α1 (X 1 )p α2 (X 2 ) p αn (X n )

7 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX 7 of orthogonal polynomials p αj (X j ) in one imension. Inee, by the prouct property n p α1 (X 1 )p α2 (X 2 ) p αn (X n ), X β y = p αj (X j ), X β j j y, whence the prouct of single variable orthogonal polynomials satisfies all requirements liste in 3.2. Triangularity in one variable an the prouct form property (3.3) forces p α to have what we call a fully triangular form: j=1 (3.4) p α (X) := γ α ρ αγ X γ, α. Also note that for any γ α there is a linear functional L y of prouct type making ρ αγ not zero. In orer to construct such a functional we consier first the Laguerre polynomials in one variable L (σ), that is L (σ) k (x) = ex x σ k k! x k (e x x n+σ ) = k j=0 k ( k + σ k j ) ( x) j. j! These polynomials satisfy an orthogonality conition on the semi-axis R +, with respect to the weight e x x σ, see for instance [4]. Remark that the egree in σ of the coefficient of x j is precisely k j. Thus the functional L y (p) = p(x 1,..., x n )e x 1... x n x σ xσn n x 1...x n, p R[X], R + n will have the property, that is the associate (Laguerre σ ) orthogonal polynomials are Λ α (X) = n j=1 L(σ j) α j (X j ). We formalize a simple observation as a lemma because we use it later. Lemma 2. The coefficients ρ α,β in the ecomposition Λ α (X) = β α ρ α,β X β are polynomials in σ = (σ 1,..., σ n ), viewe as inepenent variables, an the multi-egree of ρ α,β (σ) is α β. Note that for an appropriate choice of the parameters σ j, 1 j n, the coefficients ρ α,β in the ecomposition Λ α (X) = β α ρ α,β X β are linearly inepenent over the rational fiel, an hence non-null. To prove this evaluate σ on an n-tuple of algebraically inepenent transcenental real numbers over the rational fiel.

8 8 HELTON, LASSERRE, AND PUTINAR 3.2. Proof of Theorem 1. Proof. Let L y be a linear functional for which M (y) 0 an let (p α ) enote the family of orthogonal polynomials with respect to L y. Orthogonality in (3.2) for expansions (3.4) reas δ αβ = p α, p β y = ρ αγ ρ βσ X γ, X σ y. In matrix notation this is just γ α, σ β I = D M (y) D T where D is the matrix D = (ρ αγ ) α, γ. Its columns are inexe (as before) by monomials arrange in GLex orer, an likewise for its rows. That ρ αγ = 0 if γ α, implies that ρ αγ = 0 if α < gl γ which says precisely that D is lower triangular. Moreover, its iagonal entries ρ ββ are not 0, since p β must have X β as its highest orer term. Because of this an triangularity, D is invertible. Write M (y) = D 1 (D T ) 1 an M (y) 1 = D T D. Our goal is to etermine which entries of M (y) 1 are force to be zero an we procee by writing the formula Z := M (y) 1 = D T D as z αβ = (3.5) ρ γα ρ γβ = ρ γα ρ γβ = γ max(α,β) γ, γ β γ, α γ, γ ρ γα ρ γβ We emphasize (since it arises later) that this uses only the full triangularity of the orthogonal polynomials rather than that they are proucts of one variable polynomials. If full triangularity were replace by triangularity w. r. to < gl, then the first two equalities in (3.5) woul be the same except that β γ, α γ, γ woul be replace by β gl γ, α gl γ, γ. To continue with our proof, consier (α, β) an set η := max(α, β). If max(α, β) >, then z α,β = 0, since the sum in equation (3.5) is empty. This is the forwar sie of Theorem 1. Conversely, when max(α, β) the entry z αβ is a sum of one or more proucts ρ γα ρ γβ an so is a polynomial in σ. If this polynomial is not ientically zero, then some value of σ makes z αβ 0, so (α, β) is not a congenital zero. Now we set out to show that z αβ as a polynomial in σ is not ientically 0. Lemma 2 tells us each prouct ρ γ,α ρ γ,β is a polynomial whose multiegree in σ is exactly 2γ α β. The multi-inex γ is subject to the constraints max(α, β) γ an γ. We fix an inex, say j = 1 an choose ˆγ = max(α, β) + ( max(α, β), 0,..., 0).

9 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX 9 Note the prouct ρˆγα ρˆγβ is inclue in the sum (3.5) for z α,β an it is a polynomial of egree 2 2 max(α, β) + 2 max(α 1, β 1 ) α 1 β 1 with respect to σ 1. By the extremality of our construction of ˆγ, every other term ρ γα ρ γβ in z αβ will have smaller σ 1 egree. Hence ρˆγα ρˆγβ can not be cancelle, proving that z αβ is not the zero polynomial. 4. Partial inepenence In this section we consier the case where only a partial inepenence property hols. We ecompose the variables into isjoint sets X = (X(1),..., X(k)) where X(1) = (X(1) 1,..., X(1) 1 ), an so on. The linear functional L y is sai to satisfy a partial inepenence property (w.r. to the fixe grouping of variables), if L y (p 1 (X(1))...p k (X(k))) = k L y (p j (X(j)), where p j is a polynomial in the variables from the set X(j), respectively. Denote by eg X(j) Q(X) the egree of a polynomial Q in the variables X(j). Assuming that L y is a positive functional, one can associate in a unique way the orthogonal polynomials p α, α N n. Note that the lexicographic orer on N n respects the grouping of variables, in the sense j=1 (α 1,..., α k ) < gl (β 1,..., β k ) if an only if, either α 1 < gl β 1, or, if α 1 = β 1, then, either α 2 < gl β 2, an so on. Let α = (α 1,..., α k ) be a multi-inex ecompose with respect to the groupings X(1),..., X(k). Then, the uniqueness property of the orthonormal polynomials implies p α (X) = p α1 (X(1))...p αk (X(k)), where p αj (X(j)) are orthonormal polynomials epening solely on X(j), an arrange in lexicographic orer within this group of variables. Theorem 3. Let L y be a positive linear functional satisfying a partial inepenence property with respect to the groups of variables X = (X(1),..., X(k)). Let α = (α 1,..., α k ), β = (β 1,..., β k ) be two multi-inices ecompose accoring to the fixe groups of variables, an satisfying α, β. Then the (α, β)-entry in the matrix M (y) 1 is congenitally zero if an only if for every γ = (γ 1,..., γ k ) satisfying γ j gl α j, β j, 1 j k, we have γ >. The proof will repeat that of Theorem 1, with the only observation that p α (X) = γ. c α,γx 1 j k γ j gl α j

10 10 HELTON, LASSERRE, AND PUTINAR 5. Full triangularity A secon look at the proof of Theorem 1 reveals that the only property of the multivariate orthogonal polynomial we have use was the full triangularity form (3.4). In this section we provie an example of a nonprouct measure which has orthogonal polynomials in full triangular form, an, on the other han, we prove that the zero pattern appearing in our main result, in the inverse moment matrices M r (y) 1, r, implies the full triangular form of the associate orthogonal polynomials. Therefore, zeros in the inverse M 1 are coming from some triangularity property of orthogonal polynomials rather than from a prouct form of M. Example 1. We work in two real variables (x, y), with the measure µ = (1 x 2 y 2 ) t xy, restricte to the unit isk x 2 + y 2 < 1, where t > 1 is a parameter. Let P k (u; s) enote the orthonormalize Jacobi polynomials, that is the univariate orthogonal polynomials on the segment [ 1, 1], with respect to the measure (1 u 2 ) s u, with s > 1. Accoring to [6, Example 1, Chapter X], the orthonormal polynomials associate to the measure µ on the unit isk are: Q m+n,n (x, y) = P m (x, t + n + 1/2)(1 x 2 ) n/2 y P n ( ; t) 1 x 2 an Q n,m+n (x, y) = P m (y, t + n + 1/2)(1 y 2 ) n/2 x P n ( ; t). 1 y 2 Remark that these polynomials have full triangular form, yet the generating measure is not a prouct of measures. Theorem 4 (Full triangularity theorem). Let y = (y α ) α N n be a multisequence, such that the associate Hankel forms M (y) are positive efinite, where is a fixe positive integer. Then the following hols. For every r, the (α, β)-entry in M r (y) 1 is 0 whenever max(α, β) > r if an only if the associate orthogonal polynomials P α, α, have full triangular form. Proof. The proof of the forwar sie is exactly the same as in the proof of Theorem 1. The converse proof begins by expaning each orthogonal polynomial p α as (5.1) p α (X) := ρ αγ X γ, α γ gl α where we emphasize that γ gl α is use as oppose to (3.4) an we want to prove that β gl α an β α implies ρ αβ = 0. Note that when α = the inequality (5.2) β α is equivalent to max(α, β) >.

11 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX 11 The first step is to use the zero locations of M (y) 1 to prove that (5.3) ρ αβ = 0 if max(α, β) >, only for α =. Once this is establishe, then we apply the same argument to prove (5.2) for α = where is smaller. If n = 1 there is nothing to prove. Assume n > 1 an ecompose N n as N N n 1. The corresponing inices will be enote by (i, α), i N, α N n 1. We shall prove (5.3) by escening inuction on the grae lexicographic orer applie to the inex (i, α) the following statement: Let i + α = an assume that (j, β) < gl (i, α) an (j, β) (i, α). Then ρ (i,α),(j,β) = 0. The precise statement we shall use is equivalent (because of (5.2)) to: The inuction hypothesis: Suppose that (5.4) ρ (i,α ),(j,β) = 0 if max(i, j) + max(α, β) >, hols for all inices (i, α ) > gl (i, α), with i + α = i + α =. We want to prove ρ (i,α),(j,β) = 0. Since we shall procee by inuction from the top, we assume that i + α = that (j, β) (i, α) that (j, β) < gl (i, α) an let (i, α) enote the largest such w. r. to < gl orer. Clearly, i =. There is only one corresponing term in the grae lexicographic sequence of inices of length less than or equal to, namely (, 0). We shall prove that for every β N n 1, (j, β) gl (, 0), β > 0, we have ρ (,0),(j,β) = 0. Since the corresponing entry in M (y) 1, enote henceforth as before by z : z (,0),(j,β) = 0, is zero by assumption, an because we obtain ρ (,0),(j,β) = 0. z (,0),(j,β) = ρ (,0),(,0) ρ (,0),(j,β) Now we turn to proving the main inuction step. Assuming (5.4), we want to prove ρ (i,α),(j,β) = 0. Let (j, β) gl (i, α) subject to the conition max(i, j) + max(α, β) >. Then by hypothesis 0 = z (i,α),(j,β), so the GLex version of expansion (3.5) gives 0 = ρ (i,α),(i,α) ρ (i,α),(j,β) + + i >i, i + α = i =i, α > gl α, i + α = ρ (i,α ),(i,α)ρ (i,α ),(j,β). ρ (i,α ),(i,α)ρ (i,α ),(j,β)

12 12 HELTON, LASSERRE, AND PUTINAR We will prove that the two summations above are zero. Inee, if i > i, then max(i, i ) + max(α, α) > i + α =, an the inuction hypothesis implies ρ (i,α ),(i,α) = 0. Which eliminates the secon sum. Assume i = i, so that α = α. Then if max(α, α ) equals to either α or α, we get α = α, but this can not be, since α > gl α. Thus i+ max(α, α ) > an the inuction hypothesis yiels in this case ρ (i,α ),(i,α) = 0. Which eliminates the first sum. In conclusion ρ (i,α),(i,α) ρ (i,α),(j,β) = 0, which implies ρ (i,α),(j,β) = 0, as esire. Our inuction hypothesis is vali an we initialize it successfully, so we obtain the working hypothesis: ρ (i,α),(j,β) = 0, whenever (j, β) < gl (i, α), (i, α) =, an max( (j, β), (i, α) ) >. By (5.2) this is full triangularity uner the assumption i + α =. In contrast to the above full triangularity criteria, the prouct ecomposition of a potential truncate moment sequence (y α ) αi can be ecie by elementary linear algebra. Assume for simplicity that n = 2, an write the corresponing inices as α = (i, j) N 2. Then there are numerical sequences (u i ) i an (v j ) j with the property if an only if y (i,j) = u i v j, 0 i, j, rank(y (i,j) ) i,j=0 1. A similar rank conition, for a corresponing multilinear map, can be euce for arbitrary n. 6. Conitional Triangularity The aim of this section is to exten the full triangularity theorem of 5, in a more general context Conitional triangularity. In this new setting we consier two tuples of variables X = (x 1,..., x n ), Y = (y 1,..., y m ), an we will impose on the concatenate tuple (X, Y ) the full triangularity only with respect to the set of variables Y. The conclusion is as expecte: this assumption will reflect the appearance of some zeros in the inverse M 1 of the associate truncate moment matrix. The proof below is quite similar

13 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX 13 to that of Theorem 4 an we inicate only sufficiently many etails to make clear the ifferences. We enote the set of inices by (α, β), with α N n, β N m, equippe with the grae lexicographic orer < gl. In aition, the set of inices β N m which refers to the set of variables Y, is also equippe with the full grae orer. Let y = (y α ) α N n be a multi-sequence, such that the associate Hankel forms M (y) are positive efinite, where is a positive integer. As before we enote: p (α,β) (X, Y ) = ρ (α,β),(α,β )X α Y β (α,β) gl (α,β ) the associate orthogonal polynomials, an by z (α,β),(α,β ) = ρ (γ,σ),(α,β) ρ (γ,σ),(α,β ) the entries in M (y) 1. (γ,σ) gl (α,β),(α,β ) Definition 2. (Conitional triangularity) The orthonormal polynomials p α,β R[X, Y ], with α + β 2, satisfy the conitional triangularity with respect to X, if when X is fixe an consiere as a parameter, the resulting family enote {p α,β X} R[Y ] is in full triangular form with respect to the Y variables. More precisely, the following (O) conition below hols. (O) : [ (α, β ) gl (α, β), (α, β), an β β ] ρ (α,β),(α,β ) = 0 Next, for a fixe egree 1 we will have to consier the following zero in the inverse conition Definition 3. (zero in the inverse conition (V ) ) Let (α, β ) gl (α, β) with (α, β), be fixe, arbitrary. If (γ, σ) > whenever (γ, σ) gl (α, β), (α, β ) an σ max(β, β )), then z (α,β),(α,β ) = 0. The main result of this paper asserts that both conitions (V ) an (O) are in fact equivalent. Theorem 5 (Conitional triangularity). Let y = (y α ) α N n be a multisequence an let be an integer such that the associate Hankel form M (y) is positive efinite. Then the zero in the inverse conition (V ) r, r, hols if an only if (O) hols, i.e., if an only if the orthonormal polynomials satisfy the conitional triangularity with respect to X. Proof. Clearly, from its efinition, (O) implies (O) r for all r. One irection is obvious: Let r be fixe, arbitrary. If conition (O) r hols, then a pair ((α, β ) gl (α, β)) subject to the assumptions in (V ) r will leave not a single term in the sum giving z (α,β),(α,β ).

14 14 HELTON, LASSERRE, AND PUTINAR Conversely, assume that (V ) hols. We will prove the vanishing statement (O) by escening inuction with respect to the grae lexicographical orer. To this aim, we label all inices (α, β), (α, β) = in ecreasing grae lexicographic orer: (α 0, β 0 ) > gl (α 1, β 1 ) > gl (α 2, β 2 ).... In particular = α 0 α 1... an 0 = β 0 β 1... To initialize the inuction, consier (α, β ) gl (α 0, β 0 ) = (α 0, 0), with β 0, that is β 0. Then 0 = z (α0,β 0 ),(α,β ) = ρ (α0,β 0 ),(α 0,β 0 )ρ (α0,β 0 ),(α,β ). Since the leaing coefficient ρ (α0,β 0 ),(α 0,β 0 ) in the orthogonal polynomial is non-zero, we infer [(α, β ) < gl (α 0, β 0 ), β β 0 ] ρ (α0,β 0 ),(α,β ) = 0, which is exactly conition (O) applie to this particular choice of inices. Assume that (O) hols for all (α j, β j ), j < k. Let (α, β ) < gl (α k, β k ) with β β k, that is, max(β, β k ) > β k. In view of (V ), k 1 z (αk,β k ),(α,β ) = ρ (αk,β k ),(α k,β k )ρ (αk,β k ),(α,β ) + ρ (αj,β j ),(α k,β k )ρ (αj,β j ),(α,β ). Note that the inuction hypothesis implies that for every 0 j k 1, at least one factor ρ (αj,β j ),(α k,β k ) or ρ (αj,β j ),(α,β ) vanishes. Thus, j=0 0 = z (αk,β k ),(α,β ) = ρ (αk,β k ),(α k,β k )ρ (αk,β k ),(α,β ), whence ρ (αk,β k ),(α,β ) = 0. Once we have exhauste by the above inuction all inices of length, we procee similarly to those of length 1, using now the zero in the inverse property (V ) 1, an so on. Remark that the orering (X, Y ) with the tuple of full triangular variables Y on the secon entry is important. A low egree example will be consiere in the last section. We call Theorem 5 the conitional triangularity theorem because when the variables X are fixe (an so can be consiere as parameters), then the orthogonal polynomials now consiere as elements of R[Y ], are in full triangular form, rephrase as in triangular form conitional to X is fixe The link with partial correlation. Let us specialize to the case = 1. Assume that the unerlying joint istribution on the ranom vector X = (X 1,..., X n ) is centere, that is, X i µ = 0 for all i = 1,..., n; then M reas M = 1 0 an M 1 = R 0 R 1

15 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX 15 where R is just the so-calle covariance matrix. Partitionning the ranom vector in (Y, X i, X j ) with Y = (X k ) k i,j, yiels R = var (Y ) cov (Y, X i) cov (Y, X j ) cov (Y, X i ) var (X i ) cov (X i, X j ), cov (Y, X j ) cov (X i, X j ) var (X j ) where var an cov have obvious meanings. The partial covariance of X i an X j given Y, enote cov (X i, X j Y ) in Wittaker [7, p. 135], satisfies: (6.1) cov (X i, X j Y ) := cov (X i, X j ) cov (Y, X i )var (Y ) 1 cov (Y, X j ). After scaling R 1 to have a unit iagonal, the partial correlation between X i an X j (partialle on Y ) is the negative of cov (X i, X j Y ), an as alreay mentione, R 1 (i, j) = 0 if an only if X i an X j have zero partial correlation, i.e., cov (X i, X j Y ) = 0. See e.g. Wittaker [7, Cor an 5.8.4]. Corollary 6. Let = 1. Then R 1 (i, j) = 0 if an only if the orthonormal polynomials of egree up to 2, associate with M 1, satisfy the conitional triangularity with respect to X = (X k ) k i,j. Proof. To recast the problem in the framework of 6.1, let Y = (X i, X j ) an rename X := (X k ) k i,j. In view of Definition 3 with = 1, we only nee consier pairs (α, β ) gl (α, β) with α = α = 0 an β = (0, 1), β = (1, 0). But then σ max[β, β ] = (1, 1) implies (γ, σ) 2 >, an so as R 1 (i, j) = 0, the zero in the inverse conition (V ) hols. Equivalently, by Theorem 5, (O) hols. Corollary 6 states that the pair (X i, X j ) has zero partial correlation if an only if the orthonormal polynomials up to egree 2, satisfy the conitional triangularity with respect to X = (X k ) k i,j. That is, partial correlation an conitional triangularity are equivalent. Example 2. Let = 1, an consier the case of three ranom variables (X, Y, Z) with (centere) joint istribution µ. Then suppose that the orthonormal polynomials up to egree = 1 satisfy the conitional triangularity property (V ) 1 w.r.t. X. That is, p 000 = 1 an p 100 = α 1 + β 1 X; p 010 = α 2 + β 2 X + γ 2 Y ; p 001 = α 3 + β 3 X + γ 3 Z, for some coefficients (α i β i γ i ). Notice that because of (O) 1, we cannot have a linear term in Y in p 001. Orthogonality yiels that X γ, p α = 0 for all γ < gl α, i.e., with E being the expectation w.r.t. µ, β 2 E(X 2 ) + γ 2 E(XY ) = 0 β 3 E(X 2 ) + γ 3 E(XZ) = 0 β 3 E(XY ) + γ 3 E(Y Z) = 0. Stating that the eterminant of the last two linear equations in (β 3, γ 3 ) is zero yiels E(Y Z) E(X, Y )E(X 2 ) 1 E(X, Z) = 0,

16 16 HELTON, LASSERRE, AND PUTINAR which is just (6.1), i.e., the zero partial correlation conition, up to a multiplicative constant. This immeiately raises two questions: (i) What are the istributions for which the orthonormal polynomials up to egree 2 satisfy the conitional triangularity with respect to a given pair (X i, X j )? (ii) Among such istributions, what are those for which conitional inepenence with respect to X = (X k i,j ) also hols? An answer to the latter woul characterize istributions for which zero partial correlation imply conitional inepenence (like for the normal istribution) Conitional inepenence an zeros in the inverse. We have alreay mentione that in general, conitional inepenence is not etecte from zero entries in the inverse of R 1 (equivalently, M1 1 ), except for the normal joint istribution, a common assumption in Graphical Gaussian Moels. Therefore, a natural question of potential interest is to search for conitions on when conitional inepenence in the non gaussian case is relate to the zero in the inverse property (V ), or equivalently, the conitional triangularity (O), not only for = 1 but also for > 1. A rather negative result in this irection is as follows. Let be fixe, arbitrary, an let M = (y ijk ) 0 be the moment matrix of an arbitrary joint istribution µ of three ranom variables (X, Y 1, Y 2 ) on R. As we are consiering only finitely many moments (up to orer 2), by Tchakaloff s theorem there exists a measure ϕ finitely supporte on, say s, points (x (l), y (l) 1, y(l) 2 ) R 3 (with associate probabilities {p l }), l = 1,..., s, an whose all moments up to orer 2 match those of µ; see e.g. Reznick [5, Theor. 7.18]. Let us efine a sequence {ϕ t } of probability measures as follows. Perturbate each point (x (l), y (l) 1, y(l) 2 ) to (x(l) + ɛ(t, l), y (l) 1, y(l) 2 ), l = 1,..., s, in such a way that no two points x (l) + ɛ(t, l) are the same, an keep the same weights {p l }. It is clear that ϕ t satisfies: 1 = Prob[Y = (y (l) 1, y(l) 2 ) X = x(l) + ɛ(t, l)] = Prob[Y 1 = y (l) 1 X = x(l) + ɛ(t, l)] Prob[Y 2 = y (l) 2 X = x(l) + ɛ(t, l)] for all l = 1,..., s. That is, conitional to X, the variables Y 1 an Y 2 are inepenent. Take a sequence with ɛ(t, l) 0 for all l, as t, an consier the moment matrix M (t) associate with ϕ t. Clearly, as t, X i Y j 1 Y k 2 ϕ t X i Y j 1 Y k 2 µ = y ijk, i, j, k : i + j + k 2, i.e., M (t) M. Therefore, if the zero in the inverse property (V ) oes not hol for M, then by a simple continuity argument, it cannot hol for any M (t) with

17 MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX 17 sufficiently large t, an still the conitional inepenence property hols for each ϕ t. One has just shown that for every fixe, one may easily construct examples of measures with the conitional inepenence property, which violate the zero in the inverse property (V ). References [1] C.F. Dunkl, Yuan Xu. Orthogonal Polynomials of Several Variables, Cambrige University Press, Cambrige, [2] C. Berg. Fibonacci numbers an orthogonal polynomials. Preprint. [3] E. Cripps, C. Carter, R. Kohn. Variable selection an covariance selection in multiregression moels. Technical report # , Statistical an Applie Mathematical Sciences Institute, Research Triangle Park, NC, [4] W. Magnus, F. Oberhettinger, Formulas an Theorems for the Functions of Mathematical Physics, Chelsea reprint, New York, [5] B. Reznick. Sums o Even Powers of real Linear Forms, Amer. Math. Soc. Memoirs vol. 96, Amer. Math. Soc. Provience, R.I., [6] P. K. Suetin, Orthogonal polynomials in two variables (in Russian), Nauka, Moscow, [7] J. Wittaker. Graphical Moels in Applie Mathematical Analysis, Wiley, New York, 1990.