ASYMPTOTIC EXPANSION OF A COMPLEX HYPERGEOMETRIC FUNCTION

Transcription

1 ASYMPTOTIC EXPANSION OF A COMPLEX HYPERGEOMETRIC FUNCTION by Aksel Bertelsen TECHNICAL REPORT No. 145 November 1988 Department ofstatistics, GN-22 University.of Washington Seattle, Washington USA

2 ASYMPTOTIC EXPANSION OF A COMPLEX HYPERGEOMETRIC FUNCTION by AKSEL BERTELSEN* Institute for Mathematical Statistics University of Copenhagen Denmark Technical Report No. /lfs Abstract A technique for obtaining asymptotic expansions for complex hypergeometric functions is demonstrated. The method is used to find an asymptotic expansion for the function 2fA. '" This work Danish ihl,igur'ttl Science Counsel,

3

4 1 asymptotic expansrons for real hypergeometric functions, for example, Muirhead (1978) and Srivastava (1980). However, only a few papers on asymptotic expansions for complex hypergeometric functions have appeared, which seems strange since the two kinds of functions formally look very much the same. One would expect that the methods used for the real functions also would work for the complex functions. This is not so, however, because certain functions appearing in the integral representations for the complex hypergeometric functions obtain their maximum on a whole subspace of IRm, and not just at isolated points. In paper it shown how it is possible to solve problem by reducing the number of variables using the lemma found in Section 3. Li et al, (1970) had to solve the same problem but their way of doing this was not explained in detail. We have chosen to consider the complex hypergeometric function 2Fl for an example; this function appears in the density for the distribution of complex canonical correlation coefficients (James (1964), (112)). Furthermore, we have chosen to apply the method used for the real function 2Fl by Glynn (1980). 2. Notations. In this paper we consider matrices with complex elements. 1i+(p, <C) denotes the set of positive definite hermitian p x p matrices, U(p) the group of unitary p x p matrices and V(p, q) the set ofqx p matrices Q for which Q'Q = I, where Q' is the conjugate transpose of Q p:::; q. The normed invariant measure on V(p,q) is called dq. vve let.rp(q)-l, p IIr(q - i i=l where A= E =II

5 2 of c11a,go:nal matrices with c11a,go:nal elements Ull-, up IS 3. Integral representation of Let pz and R 2 be diagonal matrices. pz = diag(p;) and R 2 diagctf), where 1 > PI >... > Pp > 0 and 1 > "11 >... > "Ip > O. The complex hypergeometric function 2FHN,N,qjP2,RZ) (which in the following will be called 2FU can be expressed as (1) I\(N)-z{ IABIN-Pexp(tr(-A JfJ.(l) 1 1 ) B + 2 Re([A?iB?iPQ'R : OlE)) dadbdqdf where n(l) = H+(p,<D) x H+(p,<D) x U(p) x V(p,q), and s > p (see James (1964), (86) and 91-92)). We shall derive the asymptotic behavior of zia for large N using the following: if function J(x) = I(Xl,.., X m ) has. an absolute max;imurn. Jat an interior of.a domain S ina real 'In-dimen.sional space, then under suitable conditions, as N ---t (2) log to we can expressed as

6 3 g(a, R, can a A E <lj) on the A GU 2G', where G E U(p) and U E D(U). Using (93) James (1964) we can (3) as (4) O 2 f!(u,g, V,H,P,Q,R,F)Nh(U, V)dUdGdVdHdQdF 10(2) where n(2) = D(U) x U(p) x D(V) x U(p) x U(p) X V(p,q), f(u,g, V,H,P,Q,R,F) = IUVI 2 exp (tr( _U 2 V 2 +2 Re([UGVHPQ'R : O]F))) and. v(p,p).2 2p The asymptotic behavior for the real hypergeometric function 2Fl was obtained by applying (2) to an integral of the form (4) after making a proper parametrization of the compact groups. That does not work in the complex case since! does not have an isolated In to this nrooiem we will now it is possible reduce Let U 1 (p) be the manifold consisting of matrices in U(p) with real positive diagonal eleltiejlts; let sut>.e;i'(>up of U(p) consisting can be rdentmed coset of traastermations on this normed invariant measure on For G E a unique decomposnaon E nol~m(~d li:lvalrlallt measure Lemma. a tnnctson ap p

7 4 (6) xv(p,q) Now for fixed Ql and Q'l, use the invariance of df to write (6) as where Q'l, = (~'l, ButQ'l,RQ'l, 0).. IS a q X q matrix. R from which the lemma follows. o Using the lemma three times can replace G, H, and Q in (4) by G1, HI, and Q1, all three belonging to U 1(p), to get an integral representation of'l,f\ which will be denoted 'l,l, and to which we can apply (2). 4. Asymptotic representation of 'l,f\. We are now ready to Theorem 1. Let p'l, diag(pt) and R'l, = diag("d), where 1 > PI >... > Pp > 0 and 1 >,I >... > 'v > 0, then for large N (7) where p C(N) TI(l -,ipi i=1 1 p ('ipi)-2- q + P.TI ((P7 P1)(,;,;))-1 i<j Proof. are not a unique maximum

8 5 To see this, o 0 ' any complex matrix Al iaz on the real form (9) and take into account that when a diagonal matrix is written on real form all the diagonal elements appear twice. Then maximizing over U and V it follows that the maximum value of f is (10) p e- zp IT(1-riPi)-Z i=1 and the maximum is obtained at We have to use a parametrization of the compact groups to be able to apply (2), since n(2) is not an open subset.. -I A matrix (h E U 1 (p) can be expressed as G 1 = exp(zs), where S = S. We are, however, only going to use the part 1 +is -!SZ of exp(is), so we can assume that the diagonal elements of S are O. The jacobian of this transformation has the form (This follows from Li et al (1970)). Similarly HI = exp(it) and Ql exp(ivv). Finally F E V(p,q) is parametrized by writing [F: exp(iz) ( i (Zn ZZI o )) p columns are Zn is p x p nerrmtran ZZI is a x p matrix. jacobian IS

9 6 by u = V = mag (1 - 'YiPi)-~),S T = W = 0 and Z = o. The number of variables to be integrated is m p(2p + 2q - 1). We have to find the Hessian of -loge!). Fortunately we can use the calculations done by Glynn (1980) for the corresponding Hessian related to 2Fl; to see this write (once more) any complex matrix Al + ia 2 on real form (9). We find (11) ll. = 2m2'p>~ (1-7'P,)(-Y,P;/(1-7'P,»,-P+t)' p x IT ((;ripi 'YiPi)4(1- 'YiPi)-4(1- 'YiPi)-4(p; - pj)(;r; - 'YJ)) 2 i<i Inserting in (2) and reducing the expression proves Theorem 1. 0 Without proof we state the following theorem. The steps of the proof are exactly the same as for Theorem 1. Theorem 2. Let p2 diag(p7) and R 2 = diag('y7), where 1 > Pi >... Pk > Pk+l... = PP = 0 and 1 > 'Yl >... > 'Yp > 0, then for large N (12) Fl. (N, N, q; P,R ) '" k 1 k den) IT(1- 'YiPi)-2N+q+P-l('YiPi)-2+ P- q. IT (pr - P])(;r; i=1 i<i k p X IT IT ((;r; = 'YJ)pD- 1 i=1 'YJ))-1 (13)

10 7 For p 1 and p = 2 it possible to calcutase the values of 2fA from the expansion in complex zonal polynomials (James (1964), (88)). Below is result obtained on a computer for some values of N, q, p, P and R. The asymptotic values of 21ft obtained from (7)-(8) are also given. p= 1, q 3, 1 = 0.45, P 0.35 N exact asymptotic p=2, q = 3, 11 = 0.3, , PI = 0.4, P2 = 0.2 N exact asymptotic

11 8 Glynn, of the densities of correlations and roots in when population parameters have arbitrary multiplicity," Tbe Annals ofstatistics, 8, [2] Hsu, L.C. (1948). "A theorem on the asymptotic behavior of a multiple integral," Duke Matb. J. 15, [3] James, A.T. (1964). "Distribution of matrix variates and latent roots derived from normal " Ann. Matb. Statist. 35, l. [4] H.C., K.S.C. and Challg, C. (1970). "Asymptotic expansions for distributions of the roots of two matrices from classical and complex gaussian populations," Ann. Math. Statist. 41, [5] Muirhead, R.J. (1978). "Latent roots and matrix variates: a review of some asymptotic results," Ann. Statist. 6, [6] Srivastava, M.S. and Gal:'ter, E.M. (19~0). "Asymptotic expansions for hypergeometric functions," in Multivariate Analysis V, North-Holland Publishing Company,