Annales de l’institut Fourier (AIF)

Laurent, Yves

Malgrange, Bernard

Cycles proches, spécialisation et ${\mathcal {D}}$-modules

Mitsumatsu, Yoshihiko

Anosov flows and non-Stein symplectic manifolds

Huebschmann, Johannes

Poisson structures on certain moduli spaces for bundles on a surface

Ronga, Felice

Un procédé d'élimination effective et quelques applications

Sauzin, David

Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé

Zhang, Genkai

ANNALES DE L’INSTITUT FOURIER
GENKAI ZHANG
The asymptotics of spherical functions and the
central limit theorem on symmetric cones
Annales de l’institut Fourier, tome 45, no 2 (1995), p. 565-575
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Ann. Inst. Fourier, Grenoble
45, 2 (1995), 565-575
THE ASYMPTOTICS OF SPHERICAL FUNCTIONS
AND THE CENTRAL LIMIT THEOREM
ON SYMMETRIC CONES
by Genkai ZHANG
0. Introduction.
In multivariate statistical analysis the spaces of positive definite
matrices (real or complex) are of considerable interests. One studies the
spherical polynomials, central limit theorem on them. See [M]. A natural
generalization of those spaces is the cone of positive elements in a formally
real Jordan algebra. This class of cones includes cones of positive definite
real, complex and quaternion matrices and the forward light cone. In this
paper we will prove a kind of central limit theorem on any such symmetric
cones.
In his paper [T2] Terras obtained a central limit theorem for rotation-
invariant random variables on the space of n x n real positive definite
(symmetric) matrices in the case n = 3. This has been recently generalized
to any n by Richards [R]. In this paper we will generalize this central limit
theorem to the symmetric cone of positive elements in a Jordan algebra.
The main tools used in [R] are the Binet-Cauchy theorem and certain
coefficients in some recurrence formulas of zonal (spherical) polynomials in
[Ku]. Recently we have found some recurrence formulas on any simple and
formally real Jordan algebra [Z]. To prove our central limit theorem we will
Key words'. Spherical functions — Central limit theorem — Jordan algebra — Symmetric
spaces - Heat equation - Spherical polynomial.
Math. classification: 43A65 - 62E15.
566 GENKAI ZHANG
use those recurrence formulas and a recent result of Faraut and Koranyi
[FK] on expansion of reproducing kernels.
Acknowledgment. — The author would like to thank Jonathan
Arazy for comments on the paper and Bent 0rsted for constant encour-
agement.. He also thanks Donald. St. P. Richards for drawing his attention
to the work of P. Graczyk ([Gl] and [G2]). He acknowledges the financial
support from Odense University, Denmark, where this work was done.
1. Preliminaries.
In this section we recall some facts about the symmetric cone in a
Jordan algebra. Most results in this section can be found, e.g. in [UU] and
[KS] (on formally real Jordan algebras) and [Tl] (in the special case of real
symmetric matrices).
Let X be a simple and formally real Jordan algebra [BK] with product
x o y and unit element e. Basic examples are the space X = 7<y(SC) of
all self-adjoint r x r-matrices over K = R,C or El (quaternions) with
the anti-commutator product x o y = (xy + y x ) / 2 , and the "spin factor"
X = R1"^ of all vectors x = (a;o,^i,... ,^'n) = (^05 ^Q? with product
x o y = (xovo + x ' y ' , XQV' + y^x'). The open set
P = {x2^ C X invertible}
is a convex cone. For the matrix algebras, P is the cone of positive definite
matrices, whereas for the spin factor P is the forward light cone. The linear
automorphism group
GL(P)={geGL(X):g(P)=P}
is transitive on P. Let G be the identity component of GL(P). G has Lie
algebra
g = {M e gW : exp(tM) € GL(P), for all t e R}
with the commutator bracket [M, N] = MN — NM. The Lie algebra g
has a Cartan decomposition g = k4- p, where kis the Lie algebra of the
"orthogonal" group
K == 0(P) = {g € G : ge == e}
and p = {MX : x C X} consists of all multiplication operators
M^y=xoy
THE ASYMPTOTICS OF SPHERICAL FUNCTIONS 567
on X. Moreover P = G/K is now a Riemannian symmetric space.
Let e i , . . . , Or be a maximal orthogonal system of idempotents in X.
Then we have
e = ei + ... +67..
Let
(1.1) X= ^ ^
l^<j<r
be the Peirce decomposition induced by { e i , . . . , 67.}. Here
X^ = X,, = [x C X : e, ox = ̂ ^
are the joint eigenspaces of the multiplication operators {Me.Y^i on Jsr-
The integer r is independent of the choice of the maximal orthogonal system
of idempotents and is called the rank of X.
Let
a = RMei + ... 4- MMe,
be the subspace of p generated by M^,..., M^. Then a is maximal abelian
in p. So r is the rank of the Riemannian symmetric space P. We have thus
a root space decomposition of (g, a) [HI]. We let {7j}^i be the dual basis
in a* of{MeJ^i, i.e.
7j(^eJ=^fc.
Then the corresponding root system consists of 7J ~ 7fc, j ^ k with
common multiplicity a. The Weyl group is the symmetric group 67.. For
example in the case of matrix algebra 7<r(K) the rank is r and the
multiplicity is a = 1,2 or 4 according to K = R, C, or HI respectively.
We fix an ordering of a* by letting
7l < 72 < • • • < 7r.
/"y . __ '"VL. "̂
Therefore, the positive roots are 3 , j > k. Let p = ^ p^ be, as
2 ~ .7=1
usual, the half sum of positive roots
q ^._^ g_ . ^1
^ 2^ 2 2^V 2 y7^'j>fc j=i
We will now describe the spherical functions and the heat equation on
the symmetric space P. We need the notion of Jordan algebra determinants.
568 GENKAI ZHANG
Let P(x) := 2(Ma;)2—Ma;2. The determinant of it is up to a constant a
unique (integer) power of an irreducible polynomial A (a;) of degree r on X,
normalized so that A(e) = 1. A (a?) is called the determinant of the Jordan
algebra X. It can also be defined as the unique irreducible polynomial given
by the "Cramer's rule":
_i cofactorofrc , „. .. . .
x = ——————, [x € Ainvertible)
A(a;)
normalized so that A(e) = 1.
Let Xij be given in the Peirce decomposition (1.1) we put
X, = ̂  X^e
M<J
and let Pj be the orthogonal projection of X on Xj; we define
A,(aO=Ax,(P,aO
where ^x- is the corresponding determinant in Xj.
For any t = (ti , . . . , tr) € C7" we define
(1.2) A^^n^w
J=l
r
Let A = S ̂ j7j ^ ^5 ^^e complexification of a*, we let s =
j=i
(^i, S 2 , . . . , 5y.) be defined by
(1.3) Sj = \j 4- pj - Xj-^-i - pj+i,
where we put Sr = Ay. + p r ' The spherical function h\ is then
h),(x)= { ^{kx)dk= ( \\^(kx)dk.
J K ^j^i
Then h\(x) are J^-invariant eigenfunctions of all the G-invariant dif-
ferential operators on P. The Helgason-Fourier transform of a J^-invariant
function / on P is defined by
fW = / f{x)h^x)d-x^Jr
[x)ti\(x)d x,
iv
where d* is the G-invariant measure on P.
For later use we also introduce the spherical polynomials, which is
the analytic continuation of h\ (in A). Let m = (mi, . . . , rrir) be a r-tuple
of integers with m\ > m^ > ... > mr > 0. The functions
<M^)= / n^"^1^)^
^3=1
THE ASYMPTOTICS OF SPHERICAL FUNCTIONS 569
where m^-n = 0, are the spherical polynomials on P. In particular if we
put
î E^.
k=l
the functions
<M^)= { Aij(Jb;)dfc= I ^j(kx)dk
JK ~ JK
are called the fundamental spherical polynomials on P.
Finally we recall the heat equation on P. Let L be the Laplace-
Beltrami operator on P. We normalize the J^-invariant norm on p so that
norm M^ has norm 1. Then Mei, . . . , M^ are orthonormal basis of a. This
also gives a normalization of the Riemannian metric on the cone P = G / K .
Then (f)\ are eigenfunctions of L:
^'(i^-^)^.
j'=i
The heat equation is defined
f Qtu{x, t) = Lu(x, t), (x, t) C P x R+
\u{x^)=f{x).
2. The main theorem and the proof.
Before stating our main theorem we introduce some notation. Let
r
H = ^ hjMej € a. We let J(expH) be the function defined on A by
j=i
j(exp^) = —n^1 -e^Q-n^01^^.
2V2/ 2>J 1=1
Our main theorem is the following central limit theorem.
THEOREM 1. — Let Ym, m >_ 1 be a sequence of independent,
identical distributed, K-invariant random variables in P, with common
density function f(Y). Assume that f(Y) satisfies
( hif(exp H)J{exp H)dh^ • • • dhr = 0
JQL
570 GENKAI ZHANG
/ hihjf(exp H)J(exp H)dhi' • • dhr = 0, i -^ j
Jo.
and the convolution Sm = YI o Y^ o • - - o Ym is normalized as in (3.39) of
[Tl] and that Sm has density function f^. Then for any measurable subset
S c P
/^(y)d*(y)^exp^^x t G_^^F^,(Y)d\Y)
JS 9D(2 + ar) JS 2(2+ar) 2+ar
as m —> oo.
In the above, Gf is the fundamental solution of the heat equation on
P with Helgason-Fourier transform
s-.m^(^-^)).
\ i=l /
Further Fi has Helgason-Fourier transform
I r \
Ft(\)=exp t( ^ AzA,) .
\ l<fc<j^r /
Following [Tl] and [R] we need to find the expansion of the spherical
functions h\(expH) on a around H ^= 0. Recall that spherical functions
h\(expH) are Weyl group invariant in A and H G a. The expansion is
r
(recalling that H = ̂  hjMej)
j=i
r r
h^(expH) =1 + (an ̂ A, + 012) (^^z)
i=l %=1
r r r
+ (a2i^A? + 022 ^ A,A^ + a23^A, + 024} (i^)
z==l l^i<j<r 1=1 i=l
4- -P(Ai, . . . , Ay.) ( Y^ ^z/^j) + higher terms;
Ki<j<r
namely we need to find the six coefficients an, 0:12, Q;2i, 0^221 <^23) o'24-
We fix X 6 Q^ and 5 as in (1.3). Consider the formula (1.2) in the
polar coordinates
r
(2.1) A^(fca.e)=I'[A^(te.e)
j'=i
and put a = expH. Following [R] we take logarithm differentiation to
obtain
1 f) r 1 Q
(2.2) ————-——As(fca-e) =Y^5,————-——^j(ka'e).v / A^(fca • e) Qh\ — ' L^ Aj (ka - e) Qh\ J l
THE ASYMPTOTICS OF SPHERICAL FUNCTIONS 571
Integrating over K and putting H = 0 we get
-h\{a'e)(2.3) ^
^/M=E5 ll(a•e)<9/ii j^=o j'=i Jf=o
and
We adopt the notation [R] to denote
^(fc)=^-A,(fca.e)
/;W=^A,(fca.e)
f(/) = / /(^
JK
H=0
H=0
for a function / on K.
We need a result of Faraut and Koranyi [FK].
LEMMA 1. — For a = expH we have
(2.4) f[(l - te^) = ̂ (-1^ (^Ma. e)t^
j=i j=o VJ/
Proof. — This is a special case of Theorem 3.8 in [FK]. See also
Remark 2 after the theorem. We give here a simple independent proof. We
know that the polynomial A(e — z) is a polynomial of degree r on X and
is K invariant. Thus we have an expansion
r
A(e-^)=^c,^(^',
j=o
for some constants cj. Now take z = a ' e the above equation reads
n(i-^)=^c^(^'.
J=l j=0
Evaluating the equation at H = 0 and noting that <^(e) = 1 we find the
coefficients Cj. D
Lemma 1 has also been proved previously by J. Arazy and Z. Yan [A].
Now we differentiate the expansion in Lemma 1 and set H = 0,
-t(i-.r-,S(-i).(;:;)̂ (..^ij(a-e) t3'.
H=0
572 GENKAIZHANG
however the LHS of the above is
-((i-o-^a-1)^;:;)^'
j=o v /
thus we find
f(/0=——<Ma-e) = j / r .
<^1 — H=0
Using this we see
Oill = -, Oi2 =0.
V
Note that if we differentiate (2.4) twice we get
W)=|^<Mo-e)
H=0
= j / r .
Now the coefficients of 2^ h2 is
1=1
a2
(2.5)
We have
Q(5i , . . . , s^)= -^h^a-e)
H=O
Q(.i,...,^)=<p((^y^)')+^(^(/;)-^(^^^
j'=i j=i
We need to find
W^ W)2)-
We start to calculate £{f[f'j)- It follows from §1 that
Ai^+H = AiAj.
Differentiating at H = 0 and integrating we find
(2.6) -i-^^ • e) = ̂ (^0 + <?(/;) + 2^ (/{')<?(/;).
^"'i ^=o
To find out the LHS we need the a result proved in [Z] (see also [St]).
LEMMA 2. — We have the recurrence formula of the product 0ii<^ij
of the spherical polynomials
î= ̂ T^)^^ + ̂ r^T)^
for the product (̂ 11 (^u.
THE ASYMPTOTICS OF SPHERICAL FUNCTIONS 573
The lemma can also be written as
l̂±lrL A. A. r - ^ .
rdT^)^^ = ̂ ^ - rO-TtJ)01^'
Thus
^^t-'„„=(•+jJ)(2j+^:)|,(r-J)u+•)
Now we substitute this to (2.6) to get
In particular
^'OT
W)'^-
Thus the coefficient of AiA2 m (2.5) is
2a22=2(f((/^)-£((,if((/ )- ((/02))
2a
2r + ar2
and the coefficient of A^ is
2a2i=£((/{)2)
_ 2+Q
~~ 2r + or2'
The coefficient of Ai in (2.5) is
2023 = £(f[) - £ ((/Q2) - a^f(/^) + ^(r - l)f(/{)
J=2 z
=0.
On the other hand, we have Q(si,... ,Sr) = 0 for s = 0 or for
^ = -|(^ - 7-2—)' that is
r
0 = Q'21 ̂  A^ + 0:22 ^ A^A^ 4- 0:24
j=l Ki<j<r
which gives us
a(r2 - 1)
Q'24 = -48(2 + ar)'
574 GENKAI ZHANG
Finally we get the required expansion
^(exp^^+^^A^^^)
1=1 %=i
( 2+0 "̂"̂  ,2 CL -̂-\+ 2(27^^or2)^ ^+2r^^ar2 2^ A^
v / %=1 l<z<7<r
_a(r2zlL\(\-h2}48(24-ar)A2^
+ P(Ai, . . . , Ay.) ( Y^ ft^j) + higher terms.
<i<j<r
The rest of the proof is similar to that in [Tl].
BIBLIOGRAPHY
[A] J. ARAZY, Personal communication.
[BK] H. BRAUN and M. KOECHER, Jordan-Algebren, Springer, Berlin, Heidelberg, New
York, 1966.
[FK] J. FARAUT and A. KORANYI, Function spaces and reproducing kernels on bounded
symmetric domains, J. Func. Anal., 89 (1990), 64-89.
[Gl] P. GRACZYK, A central limit theorem on the space of positive definite symmetric
matrices, Ann. Inst. Fourier, 42-1,2 (1992), 857-874.
[G2] P. GRACZYK, Dispersions and a central limit theorem, preprint.
[HI] S. HELGASON, Differential Geometry, Lie Groups and Symmetric Spaces, Academic
Press, London, 1978.
[H2] S. HELGASON, Groups and geometric analysis, Academic Press, London, 1984.
[KS] B. KOSTANT and S. SAHI, The Capelli Identity, Tube Domains, and the Generalized
Laplace Transform, Adv. Math., 87 (1991), 71-92.
[Ku] H. B. KUSHNER, The linearization of the product of two zonal polynomials, SIAM
J. Math. Anal., 19 (1988), 686-717.
[L] 0. LOOS, Bounded Symmetric Domains and Jordan Pairs, University of California,
Irvine, 1977.
[M] R. J. MURIHEAD, Aspects of multivariate statistical theory, John Wiley & Sons,
New-York, 1982.
[R] D. St. P. RICHARDS, The central limit theorem on spaces of positive definite
matrices, J. Multivariate Anal., 23 (1987), 13-36.
[St] R. P. STANLEY, Some Combinatorial Properties of Jack Symmetric Functions, Adv.
Math., 77 (1989), 76-115.
[Tl] A. TERRAS, Harmonic analysis on symmetric spaces and applications, II, Springer-
Verlag, New-York, 1985.
THE ASYMPTOTICS OF SPHERICAL FUNCTIONS 575
[T2] A. TERRAS, Asymptotics of special functions and the central limit theorem on the
space 'Pn of positive n x n matrices, J. Multivariate Anal., 23 (1987) 13—36.
[UU] A. UNTERBERGER and H. UPMEIER, Berezin transform and invariant differential
operators, preprint.
[Ul] H. UPMEIER, Jordan algebras in analysis, operator theory, and quantum mechanics,
Regional conference series in mathematics, n° 67, Amer. Math. Soc., 1987.
[U2] H. UPMEIER, Toeplitz operators on bounded symmetric domains, Tran. Amer.
Math. Soc., 280 (1983), 221-237.
[Vr] L. VRETARE, Elementary spherical functions on symmetric spaces, Math. Scand.,
39 (1976), 343-358.
[Z] G. ZHANG, Some recurrence formula for spherical polynomials on tube domains,
Trans. Amer. Math. Soc., to appear.
Manuscrit recu Ie 18 octobre 1994.
Genkai ZHANG,
School of Mathematics
University of New South Wales
Kensington, NSW 2033 (Australie).
genkai@solution.maths.unsw.edu.au


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