Statistics & Probability Letters 1 (1983) 141-145 North-Holland Publishing Company

March 1983

Representations for Eigenfunctions of Expected Value Operators in the Wishart Distribution Donald Richards

Department of Statistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27514, U.S.A.

Received April 1982; revised version received October 1982

Abstract. Recent articles by Kushner and Meisner (1980) and Kushner, Lebow and Meisner (1981) have posed the problem of characterising the 'EP ' functions f ( S ) for which E(f(S)) = hn f (~ ) for some h n ~ R, whenever the m x m matrix S has the Wishart distribution W(m, n, ~). In this article we obtain integral representations for all nonnegative EP functions. It is also shown that any bounded EP function is harmonic, and that EP polynomials may be used to approximate the functions in certain L p spaces.

Keyworak. Wishart matrix, expectation operator, homogeneous function, harmonic function, eigenfunctions, zonal polynomials.

AMS 1980 Subj. Class. Primary 62E15, 62H99; Secondary 45C05.

1. Introduction

In two recent articles, Kuslmer and Meisner (1980) and Kushner, Lebow and Meisner (1981) have formulated and treated the problem of classifying certain 'EP' functions. Specifically, a scalar-valued function f (S) of matrix argument if EP with re- spect to the Wishart distribution if

E(/( S)) = Xj (~ . ) (1)

whenever the m × m random matrix S has a Wishart distribution W(m, n, ,~) on n degrees of freedom, with covariance matrix Z. Thus, f (S) is EP if (1) holds for all positive definite symmetric Z, all n >I some no, and for some sequence of ei- genvalues (~'n) which are independent of Z.

For m ~< 2 detailed solutions are provided in Kuslmer and Meisner (!980), while Kushner et al. (1981) in the main discuss EP polynomials. It is seen there that the zonal polynomials of James

(1964) play a central role. Richards (1981) has obtained a characterisation of EP functions in terms of invariant differential operators. More re- cently, Davis (1981) has independently, and via different methods, obtained some of the main results of Kushner et al. (1981).

In this article the intent is to derive general descriptions of nonnegative EP functions. For EP polynomials, Kushner et al. (1981) show that the zonal polynomials may be used as generating func- tions, so that the results given here may be re- garded as an extension of their results.

Actually, we go one step further. If f(S) satis- fies the EP condition (1) for a fixed n, we shall say that f (S) is an n-EP function. Using results of Furstenberg (1965), we obtain representations (in Section 2) for all nonnegative n-EP and EP func- tions.

Section 3 extends certain connections between EP and harmonic functions which appeared earlier

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in Kushner and Meisner (1980). Finally, Section 4 shows that the EP polynomials rlaay be used to approximate the functions in certain L p spaces, extending some results of Kushner et al. (1981).

2. Integral representations

Throughout, we denote by GL(m) (respectively, O(m)) the group of m × m real nonsingular (re- spectively, orthogonal) matrices, and by P(m) the cone in GL(m) of positive definite symmetric matrices. We shall also have to deal with SL(m) (respectively, SO(m)) the subgroup of GL(m) (re- spectively, O(m)) consisting of the matrices with determinant unity.

Now, the n-EP criterion (1) may be written in the form

fe(m f (S )k (~ ' ' S )d l~ ( s )=h" f ( '~ ) ' Z ~ P ( m ) .

(2)

Here, k(Z, S)ISI "-(re+l)~2 is the Wishart density, and d/~(S) = ISI -(m+ ~)/2(dS) is the invariant mea- sure on P(m).

Since n is temporarily fixed, we suppress the suffix on ~,, and denote by V(k) the cone of nonnegative solutions of (2). The following lemma collects certain properties of V(k) which we shall need later.

Definition 2.1. A cone C of nonnegative functions on P(m) is translation-invariant if, for any f ~ C, the function fA(S)=f(ASA') (A ~ GL(m), S P(m)) also belongs to C.

Lemma 2.2. The cone V( X ) consists of continuous functions, is translation-invariant, and is closed in

p P(m) the topology of pointwise convergence on ..+ .

Proof. That V(X) is translation-invarant is proven in Kuslmer and Meisner (1980) and Kushner et al. (1981). Further, the infinite differentiability of the functions in V(X) may be deduced by regarding (2) as a multi-dimensional Laplace transform. In- deed, this view of (2) also implies that V(X) is closed; a detailed proof can be obtained by suita- bly modifying the proof of Proposition 9.5 in Berg and Forst (1975). []

Definition 2.3. A closed, translation-invariant cone C_c RP+ <m) is irreducible if C contains no proper, closed, translation-invariant sub-cone.

Except for m- - 1, V(h) is not irreducible. Be- fore we construct an irreducible sub-cone of V(h), recall that in Kushner et al. (1981) irreducible subspaces of EP polynomials were shown to be generated by the functions

m

f~(s)= l-IIs, I ~,-~,*,, i=1

where a = (a I . . . . . am) is an m-tuple of nonnega- tive integers, ot~ >/a 2 >/ • • • >/a m >/am+ l = 0, and S i is the i x i principal minor of S. Thus, it is reasonable to expect that the f,~'s should also gen- erate irreducible sub-cones of V(~,). To construct such a sub-cone, let °3K+(O(m)) denote the space of positive Borel measures on O(m), and

S . = ( a = ( . , . . . . . am): oti> - - ½ ( n + 1 - i ) .

1 ~< i ~< m, and

)~ = 2 , ,+ . . .+ , , . , f i F ( a i + l ( n + 1 - i ) ) } , _ , "

For any a ~ S,,, define

v(x; . ) = = , o . . ) :

'r ~ 6~]L+ (O(m) )} •

In Furstenberg's (1965) terminology, f~(S) is an irreducible O(m)-multiplier. That is,

(i) for any A ~ GL(m), fa(HASA'H') is pro- portional to f,(HiSH~) for some H l ~ O(m),

(ii) f~(HSH') - 1 for S = lm, (iii) the cone V(X; a) is an irreducible sub-cone

of v(x). Strictly speaking, the results of Furstenburg (1965) are formulated not for GL(m), but for SL(m). In any case, (i) is not difficult to establish, (ii) is trivial, and (iii) is a consequence of (i) and the fact that up to multiplication by positive constants, V(X; a) contains precisely one O(m)-invariant function.

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The converse of (iii), that every irreducible sub- cone of V(X) is a V(~,; a) is the content of Theo- rem 12.2 in Furstenberg (1965). It allows us to formulate our main result.

Theorem 2.4. For f ( S ) to be a nonnegative n-EP function with eigenvalue )kn, it is necessary and suficient that there exists a positive measure ~, concentrated on O( m ) × Sn, such that

f(S)=f f~,(HSH')d~,(H,a), S~P(m). "O(m)x$.

we find that the function f ( S ) = ISl' is n-EP for sufficiently large n.

Theorem 2.5. For f ( S) to be a nonnegative EP function with eigenvalues ()k,), n >~ no, it is neces- sary and sufficient that f ( S ) be of the form (3), where the positive measure I, has support on O( m ) ×$.

The proof is immediato. In addition, a number of results previously obtained in Richards (1981) directly follow from Theorem 2.5.

(3)

Proof. I f f ( S ) is of the form (3), it is clearly n-EP with eigenvalue X,. For the converse, let V* be the closure, in the topology of pointwise convergence, of the cone generated by all functions of the form (3). If there exists a g e V()~, ) \V*, then (g ) , the pointwise closure of the cone consisting of all nonnegative multiples of g, is clearly an irreduci- ble sub-cone of V(~,). By a remark above, ( g ) containsf. .(S) for some a e S,,, so t ha t f , (S ) e V*; a contradiction. D

Corollary 2.6. Let f ( S ) be a nontrivial nonnegative EP function. Then, for any invariant differential operator D, D f ( S ) is also EP.

Proof. The assertion is implied by the well-known result by Richards (1981) that f ~ ( S ) = H7'_11Sil ~'-',+, is an eigenfunction of every in- variant operator. []

An interesting special case is m = 1. Then, in Theorem 2.4 we have

$. = {a: a > - ½ n , h . = 2"r (a + ½n) / r (½n) ) .

The measure 1, in (3) is uniquely determined by f ( S ) . From Theorem 2.4 we shall obtain below a description of the nonnegative EP functions. Let us set S = ('] .~,,0S,.

Important examples of n-EP functions are pro- vided by James' (1964) zonal polynomials, which are indexed by m-tuples ~: = (k l, . . . . k , , ) of non- negative integers, where k~ >/k 2 >t • • • >/k,,, >/0. Choosing dp(H, ct)ffi c (dH)8 , where c i~ a suita- ble constant, (d H ) a norrnalised invariant measure on O(m), and 8, the Dirac measure at a = K, we have

C,( S)= c fo(m:,,( HSH' ) (dH).

This integral representation is due to James (1973) and has been used in Richards (1981) to derive differential equations for C.(S).

Less recondite examples may be obtained simi- larly; choosing dl ,(H, a) -- 808c0.0.....0.t ), where 80 is Dirac measure at the identity in O(m) and t e R,

Since the function ot -~ 2*F(a + ½n), a > - ½n, in- tersects any horizontal line in at most a finite number of points, the set $, is finite. We should also mention that for this case, Theorem 2.4 can be derived using the Choque t -Deny theorem (1959/60) (cf. the introduction in Furstenburg (1965)).

3. Harmonic functions

In this section we show that certain connections between harmonic and EP functions which were first established in Kushner and Meisner (1980) have appropriate generalisations.

To begin with we give a brief account of certain results on the theory of harmonic functions on SL(m) as treated in Furstenburg (1963). Further, we shall restrict our attention to the class of homo- geneous n-EP functions; for simplicity, we even assume X. = I.

Let (dH) denote the normalised Haar measure

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on SO(m), Pl(m) = P(m) ~ SL(m), and dpl (.) be the invariant measure on Pl(m).

Definition 3.1 (Furstenburg (1963, Definition 4.1)). A function g on Pl(m) is harmonic if

fs f g ( A H S H ' A ' ) k l ( S ) (dH) dp l (S) = L(m)"SO(m) =g(AA' ) , A ~ S L ( m ) , (4)

where k l (S ) dpl(S) is a probability measure on Pl(m).

The article by Furstenburg (1963) is a tour de force in the modern theory of harmonic functions. There, integral representations and partial dif- ferential equations for the harmonic functions are only two of a large number of properties devel- oped. We are particularly interested in the bounded harmonic functions; for any such g, the results of Furstenburg (1963) show that definition (4) is independent of the choice of the density kl ("), and that with A denoting the Laplace-Beltrami opera- tor on Pl(m), (4) is equivalent to Ag ffi 0.

We apply these results to the homogeneous n-EP functions, noting that the result below was initiated by formulae in Davis (1981).

Theorem 3.2. Let f ( S ) be a bounded, homogeneous n-EP function with X, = 1. Then, the restriction, g, o f f to Pl(m) satisfies Ag = O.

Proof. We must show that g satisfies (4). Since f is homogeneous, we can assume that, in (2), Z P~(m). In (2) set Z = AA', A ~ SL(m) and replace S by ASA' to obtain

f p t , , , / (ASA ' ) k (S ) d p ( S ) f f i f ( A A ' ) . (5)

Setting S = tS~, t = [SI-i/m in (5) and integrating over t reduces (5) to

g(AS~A')k t (SI) dp l (S t ) f f i g (aA ' ) . fPl(m) Here, k l (S~) dp I (St) is the distribution of I S I- t/ms where S is a Wishart matrix IV(m, n, I,,,). Replac- ing S I by HSIH', H ~ SO(m) and averaging over SO(m), we obtain (4), since the density kl(- ) is invariant under rotations. By previous remarks, we have A g --- 0. []

It is worth noting that our remarks above also imply that the n-EP functions in Theorem 3.2 remain unchanged if the kernel k(2~, S) in (2) is replaced by any other point-pair invariant func- tion k*, viz., k*(2~, S ) = k*(A~,A', ASA') for all A ~ GL(m). Further, known results on the solu- iions of Ag = 0 may be used to obtain the repre- sentation in Theorem 2.4.

4. Approximation by EP functions

So far, nothing has been said about the representa- tion of arbitrary EP functions. From the results obtained before, it is reasonable to expect that the real-valued EP functions have similar integral rep- resentations, with signed measures as their repre- senting measures.

Instead of pursuing this train of thought, we adopt another approach based on the result by Kushner et al. (1981) that any polynomial may be expressed as a linear combination of EP polynomi- Ms. We denote by dW~(S) the W(m, n, In) distri- bution.

Lemma 4.1. Let f ~ L P(dWn), 1 < p < oo. Then, for any e > 0, f can be approximated by a linear combi- nation of translates of EP polynomials to within e, in the L" norm.

Proof. The proof is almost standard. First, we can find a compactly supported continuous function g such that all i l l -g l lv < ½~. Next, there exists a polynomial P such that SUPslg(S)-P(S)l < ½~. Since dW n is a probability measure, the last esti- mate implies Ilg - Pllv < ½e. Hence, I l l - Pllv < e. But by Kushner et al. (1981), P(S) is a linear combination of translates of EP polynomials. []

Acknowledgment

It is a pleasure to acknowledge numerous stimulat- ing conversations on these topics with V. Mandrekhar and G. Kallianpur. Thanks are also due to A.W. Davis (1981) whose preprint partly motivated this work, to H.B. Kuslmer (1980) for making available his unpublished thesis, and to a referee for some helpful suggestions.

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References

Berg, C. and G. Forst (1975), Potential Theory on Locally Compact Abelian Groups, Ergebnisse der Mathematik 87 (Springer, Berlin-Heidelberg-New York).

Davis, A.W. (1981), The EP functions lie in the invariant subspaces, Unpublished manuscript, C.S.I.R.O., Adelaide.

Deny, J. (1959/60), Sur l'equation/~ = p*o de convolution, in: Sbminaire Brelot-Choquet-Deny: Th$orie du Potentiel Vol. 4, pp. 5.01-5.11.

Furstenberg, H. (1963), A Poisson formula for semi-simple Lie groups, Ann. Math. 77, 335-386.

Furstenberg, H. (1965), Translation-invariant cones of func- tions on semi-simple Lie groups, Bull. Amer. Math. Soc. 71, 271-326.

James, A.T. (1964), Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Satist. 35, 475-501.

James, A.T. (1973), The variance information manifold and the functions on it, in: P.R. Kirshnaiah, Multivariate Analysis - I I I (Academic Press, New York).

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Kushner, H.B., A. Lebow and M. Meisner (1981), Eigenfunc- tions of expected value operators in the Wishart distribu- tion, Part II, J. Multivariate Anal. 11, 418-433.

Richards, D. (1981), Applications of invariant differential oper- ators to multivariate distribution theory, Mimeo Series # 1366, Inst. of Statistics, Univ. of North Carolina.

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