The theory of compact group actions on locally compact abelian groups provides a unifying theory under which different invariance conditions studied in several
contexts by a number of statisticians are subsumed as special cases. For example,
Schoenberg’s characterization of radially symmetric characteristic functions on Iw” is
extended to this general context and the integral representations are expressed in
terms of the generalized spherical Bessel functions of Gross and Kunze. These same
Bessel functions are also used to obtain a variant of the Lkvy-Khinchine formula of
Parthasarathy, Ranga Rao, and Varadhan appropriate to invariant distributions

Gross, Kenneth I.

Richards, Donald St.P.

AbstractThe theory of compact group actions on locally compact abelian groups provides a unifying theory under which different invariance conditions studied in several contexts by a number of statisticians are subsumed as special cases. For example, Schoenberg's characterization of radially symmetric characteristic functions on Rn is extended to this general context and the integral representations are expressed in terms of the generalized spherical Bessel functions of Gross and Kunze. These same Bessel functions are also used to obtain a variant of the Lévy-Khinchine formula of Parthasarathy, Ranga Rao, and Varadhan appropriate to invariant distributions

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Compact group actions, spherical bessel functions, and invariant random variables 

JOURNAL OF MULTIVARIATE ANALYSIS 21, 128-138 ( 1987) 
Compact Group Actions, 
Spherical Bessel Functions, and 
Invariant Random Variables* 
KENNETH I. GROSS’ 
Division of Mathematical Sciences, National Science Foundation, 
1800 G Street N. W., Washington, D.C. 20550 
AND 
DONALD ST. P. RICHARDS 
Department of Statistics, llniversity of North Carolina, 
Chapel Hill, North Carolina 27514 
Communicated by G. Kallianpur 
The theory of compact group actions on locally compact abelian groups provides 
a unifying theory under which different invariance conditions studied in several 
contexts by a number of statisticians are subsumed as special cases. For example, 
Schoenberg’s characterization of radially symmetric characteristic functions on Iw” is 
extended to this general context and the integral representations are expressed in 
terms of the generalized spherical Bessel functions of Gross and Kunze. These same 
Bessel functions are also used to obtain a variant of the Lkvy-Khinchine formula of 
Parthasarathy, Ranga Rao, and Varadhan appropriate to invariant distributions. 
(1 1987 Academx Press. Inc 
0. INTRODUCTION 
The theory of compact group actions on locally compact abelian groups, 
as developed in [8], provides a general framework for the study of 
invariant distributions and the unification of a variety of different kinds of 
AMS 1980 subject classifications: Primary 60E05, 62H10, 43A80; Secondary 62E15, 33A65. 
Key words and phrases: Compact groups, spherical Bessel function, Levy-Khinchine and 
stochastic representations, quotient measure. 
* Supported in part by the National Science Foundatin. Grants MCS-8101545 and MCS- 
8403381. 
+ On leave from the University of Wyoming. 
128 
0047-259X/87 $3.00 
Copyrtght tc 1987 by Academic Press. Inc. 
All rights of reproduction in any lorm reserved. 
COMPACT GROUP ACTIONS 129 
invariance considered in the literature. For example, Schoenberg’s charac- 
terization [ 151 of radially symmetric distributions, when considered from 
our viewpoint, may be regarded as classifying the positive definite functions 
on [w” that are invariant under the natural action of the group O(n) of 
orthognal n x n matrices. In this classical situation, Schoenberg showed 
that the characteristic function of a radially symmetric distribution can be 
represented by an integral of certain Bessel functions. 
With this in mind we generalize Schoenberg’s results from the 
orthogonal action on [w” to a compact group U acting on a locally compact 
abelian topological group .Y. Our work, therefore, gives a general context 
for the investigations of Dawid [IS], Anderson and Fang [l], James [IO], 
and other statisticians who have considered specific orthogonal actions on 
Euclidean spaces as they pertain to invariant distributions. 
In the extension of Schoenberg’s results to the compact transformation 
group (U, X), the kernel for realizing the characteristic function of a U- 
invariant distribution is the spherical Bessel function for the transformation 
group (U, X) that was defined and studied by Gross and Kunze [8]. 
This spherical Bessel function also relates to other statistical problems. 
For example, it allows us to adapt the work of Parthasarathy, Ranga Rao, 
and Varadhan [ 1 l] to prove a Levy-Khinchine formula for infinitely 
divisible U-invariant characteristic functions on X. 
This paper is organized as follows. The context for the first eight chap- 
ters is the general theory of a compact topological transformation group 
(U, X) in which the compact group U acts by automorphisms on a locally 
compact abelian topological group X. In particular, in Chapters 3 and 4 we 
study the orbit space i&Y and describe the decomposition theorem for a 
U-invariant measure, a result crucial to much of what follows. Chapter 5 
and 6 deal with the spherical Bessel function and the generalized Schoen- 
berg theorem for (U, X). Our LevyyKhinchine formula is the subject of 
Chapters 7 and 8. 
The last four chapters are specialized to the case in which X is a linite- 
dimensional real inner product space and U is an orthogonal action on X. 
Chapter 9 deals with a general form of “polar coordinates” for (U, X), and 
Chapter 10 shows how the abstract theory of these orthogonal transfor- 
mation groups applies to a variety of examples including the classical con- 
text of Schoenberg; its generalization to matrix spaces considered by 
James, Richards, and others; the “rotatable” distributions of Dawid; and 
the invariance models of Anderson and Fang. The paper concludes with a 
general result in Chapters 11 and 12 on stochastic representations for U- 
invariant random variables. 
130 GROSS AND RICHARDS 
1. FOURIER ANALYSIS ON LOCALLY COMPACT ABELIAN GROUPS 
Throughout the paper X denotes a locally compact abelian (Hausdorff) 
topological group. We recall that the dual group X? consists of all con- 
tinuous homomorphisms of X into the multiplicative group of complex 
numbers of unit modulus. Then X itself becomes a locally compact abelian 
(Hausdorff) group relative to the compact-open topology. For /? 6 8 and 
.Y E X, we write (fi 1 .u) for 8(-u). Pontryagin duality identifies (X)^ with X 
through the formula (x I/?) = (/I 1 x) for x E X and b E X. 
For a finite positive Bore1 measure m on X, its Fourier transform is the 
function ti on 8 given by 
The mapping m --) GI is one-to-one. If m is a probability measure then ti is 
called its characteristic .function. We refer to [ 14, 161 for details. 
2. COMPACT ACTIONS 
Throughout, U denotes a compact group acting on X by 
automorphisms. That is to say, (U, X) is a compact transformation group. 
Thus, with u(x) denoted by UX, the following properties hold: The mapping 
(u, x) -+ ux is continuous from U x X to X, (uIuz) x = u,(u~x) for all x in X 
and U, , uz in U; and x + ux is a continuous automorphism of the group X 
for each u E U. By duality, (U, 8) is also a compact transformation group 
with the action defined by 
(~PIx)=(Pl~-‘x) (2.1) 
for (u, 8, x) tz U x Bx X. 
3. THE ORBIT SPACE AND INVARIANCE 
The orbit space for the action of U on X, denoted cr\X, consists of all 
orbits Ux = {ax: u E U}. Equipped with the quotient topology the orbit 
space is a locally compact Hausdortf space. The canonical quotient map 
rtc: x -+ Ux is open and continuous from X to V/X. 
A function f on X is called U-invariant if f(ux) =f(x) for all 
(u, x) E U x X. Equivalently, such an f is constant on orbits. This means 
that a U-invariant function f can be regarded as a function f on cr\X, 
where 
J(UL=J: (3.1) 
COMPACTGROUPACTIONS 131 
In order to construct U-invariant functions, let f be any continuous 
function f on X. Define 
(3.2) 
where du is normalized Haar measure on U. By the right invariance of 
Haar measure, f, is U-invariant. By standard measure theory, formula (3.2) 
is defined almost surely in X for f~ L’(X, m). 
Similarly, a measure m on X is U-invariant if m(z&) = m(B) for all 2.4 E U 
and Bore1 sets B in X. The following theorem shows that there is a one-to- 
one correspondence between invariant measures m on X and measures G 
on v\X. See [3, Sect. 2; 2, Sect. 3; or 8, Sect. 21. 
4. THEOREM 
Let m be a U-invariant measure on X. Then there exists a unique measure 
6 on U/X such that 
for all f e L’(X, m). 
In other words, 
~*f(x)dm(x)=~~,.IS,(n(x))~~(x(.r)). (4.2) 
The measure fi on v\X is called the quotient measure corresponding to the 
U-invariant measure m. This “decomposition” of an invariant measure is 
crucial to later results. 
5. THE SPHERICAL BESSEL FUNCTION AND SCHOENBERG'S THEOREM 
The spherical Bessel function for the transformation group (17, X) is the 
function JO on xx X defined by 
JcdB, x) = jL, (B I ux) du (5.1) 
for (j?, x) E 8x X. Note that JO is U-invariant in each variable; that is, 
JcdB, x) = Jo(uB, x) = JdB, ux) (5.2) 
for all (u,~,x)~UxRxX. 
683’?1’1-Y 
132 GROSS AND RICHARDS 
The spherical Bessel functions were defined in [S] in connection with the 
Fourier analysis of (U, X). To briefly indicate their significance, note that 
for a U-invariant finite measure m on X, 
h(B) = 1 Jo@, xl dm(-xl (5.3) x 
for all /3 E 2. Indeed, since m is U-invariant, so is k; for if u E U and /I E 2 
then 
rh(u/3)=S (uj?Ix)dm(x)=j (plu-‘x)dm(x) 
2 x 
Therefore. 
= tP I ux) du Wx) = jxJo@, x) dm(x). 
The following theorem generalizes the decomposition theorem for IL!“, first 
proved by Schoenberg [ 141. 
6. THEOREM 
Let m be a U-invariant probability measure on X and let fi be the 
corresponding quotient measure on v\X. Then S? is a probability measure, 
and 
+4/n = j JdB, x) d@44x)) 
U\,l 
for /I E 2. 
Proof: We apply Theorem 4 to (5.3). Thus, for U-invariant m, 
W?) = Ix JdP, xl dm(x) 
= s JdP, x) dfi(4x)) U\X 
COMPACTGROUPACTIONS 133 
by (5.2) and the fact that du has mass 1. All that remains is to verify, by the 
substitution f= 1 in Theorem 4, that fi is a probability measure if m is a 
probability measure. 
7. THE SPHERICAL BESSEL FUNCTION AND THE LEVY-KHINCHINE FORMULA 
We relate the spherical Bessel function for (U, X) to infinitely divisible 
probability measures. Our version of the Levy-Khinchine representation 
relies upon the work of Parthasarathy, Ranga Rao, and Varadhan [ 111, 
but in part is motivated by the variant of their theorem due to 
Gangolli [6]. 
Let X be separable and metric and p a probability measure on X. 
Following [ 111, we say p is idempotent if p * p = p * 6, for some x in X, 
where * denotes convolution and 6, is the Dirac measure at x. The 
measure pI is said to be a factor of p if p = p1 * pLz for some measure pz. 
Also, p is called infinite/y divisible if for each positive integer n, there exist 
x, in X and a measure 1, such that p = A; * 6,“, where 1; = I, * I, * . * 1, 
(n convolutions). The proof of the following Levy-Khinchine theorem is 
derived from Theorem 7.1 of [ 111 by applying the “radializing” formula, 
(3.2), above. 
8. THEOREM 
Let X be a locally compact abelian separable metric group, U be as before, 
and m be an infinitely divisible, U-invariant probability measure on X. If m 
has no idempotent factors and m is real-valued, then m has the representation 
The measure F is a-finite on X; it has finite mass outside every neighborhood 
of the identity in X; and 
s Cl -Jo@, x)1 dF(x) < ~0 x 
for all BEX. Also, D(p) is a non-negative, continuous U-invariant function 
satisfying 
@(PI + Bd + WI - P2) = 2C@v1) + @(B*)l 
134 GROSS AND RICHARDS 
9. ORTHOGONAL ACTIONS AND POLAR COORDINATES 
Suppose now that X is a finite dimensional real vector space with an 
inner product and that the elements of U act orthogonally on X; i.e., 
ljuxll = Ilxll for all (u, X) E U x X, where llj is the norm derived from the 
inner product. In this case one calls (U, X) an orthogonal transformation 
group. 
Throughout the remainder of the paper we assume (U, X) is an 
orthogonal transformation group. 
By polar coordinates for (U, X) we mean a decomposition C x C for a 
dense open subset x’ of X whose complement has Lebesgue measure zero. 
More specifically, we suppose that Z= UJ’U, for a fixed closed subgroup 
UO of U, C is an open subset of the subspace X,, of points in X fixed by UO, 
for each rE C the orbit Ur is equivalent to C, and the mapping 
(uU,, r) -+ ur is a diffeomorphism of Z x C onto x’. The points in X’ are 
called regular. 
In the presence of polar coordinates, the orbit space (except for a null 
set) can be identified with C, and the spherical Bessel function is deter- 
mined more explicitly on C. The following examples illustrate this 
phenomenon. 
10. EXAMPLES 
(A) Let X= 08” with the usual inner product (.I.) and U= O(n), the 
group of n x n orthogonal matrices. U acts on X by left matrix mul- 
tiplication, x + UX, u E U, x E X. The dual group J? may be identified with 
R” via the inner product, since every element of 2 is of the form 
exp( i < 0, .) ), 0 E KY’. Here, we have the usual polar coordinates: 
X’= {XE R”: x #O}; .E is the unit sphere, realized as the set UC,, where 
e0 = (1, O,..., 0)’ and U,=(uEU:uo,=o,}rO(n-1); and C= 
{ roO E R”: r > 0} E (0, co). Thus, for x # 0 in X, z(x) is identified with the 
number r = lixlj. 
If /I, XE R”, then 
where Q,(t) = 2’r(n/2) t~‘.l,(t), t ~0. Here, v = (n-2)/2, and .I, is the 
classical Bessel function of the first kind of order v. Putting these facts 
COMPACT GROUP ACTIONS 135 
together, we recover Schoenberg’s classical result from Theorem 6: If m is 
an O(n)-invariant probability measure on R”, then G(x) = $( l]xll), where 
and ti is a probability measure on [0, co). 
Here, we call the reader’s attention to the fact that the function Jo, 
originally defined on R” x Iw” is ultimately determined by a function on R. 
(B) For n>k, let X= lRnxk, the space of n x k real matrices, with inner 
product (x, y) = tr(x’y), where x’ denotes the transpose of x, and 
U = O(n). The group U acts “one-sidely” on X by left matrix mul- 
tiplication, x --t UX, u E U, x E X. 8 may be identified with X since the 
homomorphisms in f are of the form exp(i trace(B’x)). Here, Y= 
{x E X: x has maximal rank, k), C = C,,, is the Stiefel manifold {(r E [w” x k: 
C+C = 1, > which is realized as UCY,,, where CT~ is the n x k matrix with 
(o,),~= 1 for all j = l,..., k and all other entries are zero; U, = 
{u~U:uo,=o,}~O(n-k); and c={a,r~X: r is a kxk (symmetric) 
positive definite matrix ). Thus, for x E aB”” k such that r = x’x is invertible, 
n(x) is identified with the positive definite k x k matrix r’/‘. 
If /I, x E R” x k, then 
Joa -xl = co,,, exp(i trace(B’ux)) ~2.4 
= constant x A ,,(~x’xfi’j?), 
where A _ ,,J.) is the Bessel function of matrix argument defined by Herz 
[9]; see also [S]. The corresponding generalization of Schoenberg’s 
theorem now reads: If m is an O(n)-invariant probability measure on R” x k, 
then G(x) =&x’x), where for any t E {t 30}, the cone of symmetric 
positive definite k x k matrices, 
4(t) = constant x 
s 
A lj2(tr) dfi(r), 
r>O 
and fi is a probability measure on {r 2 0). 
We remark that Example A is the special case k = 1. Observe, again, the 
phenomenon that the function J,, originally defined for n x k matrices is 
determined by a function on the lower dimensional space of k x k matrices. 
We note in passing that random matrices having distributions of the type 
treated in this example have been studied in the work of Richards [ 131 and 
others. 
136 GROSS AND RICHARDS 
(C) Let x= 1w;y”m”, the space of n x n real symmetric matrices. The 
group U = O(n) acts “two-sidedly” on X via ,Y + uxu ‘, ME U, XE X. As in 
the previous example, it may be shown that X can be identified with X. In 
addition, v\X may be identified with the set A,, of n x n real diagonal 
matrices. For /?, .Y E X, 
JO(P? -u) = j(),,l, exp( i trace(pusu ‘,)du=,F,(iP,x), 
where ,-,F, is the hypergeometric function of two matrix arguments defined 
by James [lo]. It is straightforward to write down the generalized Schoen- 
berg theorem for this example. 
Random matrices having two-sided invariant distributions have been 
studied by Dawid [5] who calls such distributions rotatable. 
(D) Let X= X= [Wnxk and U = O(n) x .. . x O(n), the direct product of 
k copies of O(n). Let x, ,..., xk be the columns of x E X and u = (u, ,..., u,) be 
a typical element of U, where ui E O(n) for all i = I,..., k. The group U acts 
on X via x = [x, ,..., sk] -+ [u,x,, u2x2 ,..., ukxk]. This is just k copies of the 
Euclidean example A. 
If B = c/l, ,..., pk] E 2, and x = [.x, ,..., xk] E X, then 
where a,,(.) is defined in Example A. From this, we immediately derive a 
Schoenberg-type theorem obtained previously by Anderson and Fang Cl]. 
We also note that examples analogous to those above can be constructed 
for matrices with entries in any real finite-dimensional division algebra [F. 
When IF = @, the spherical Bessel functions are related to the complex 
generalized hypergeometric functions of James [lo]; see also [7, 81. 
11. STOCHASTIC REPRESENTATIONS 
In the classical situation, Schoenberg’s theorem is sometimes restated in 
terms of stochastic representations. We shall now extend this concept to 
the group setting using polar coordinates for the orthogonal transfor- 
mation group (U, X). 
COMPACT GROUP ACTIONS 137 
12. THEOREM 
Let 2 be an X-valued random variable with associated probability measure 
m. Further, assume that m is supported on the set x’ of regular elements of 
X, where 3 = OR is the unique decomposition of 3 through polar coor- 
dinates. Then m is U-invariant if and only if0 and R are independent, and 0 
is uniformly distributed with respect to the normalized U-invariant measure 
on z= u/u,,. 
Here, we write nZ(x) = CJ and nC(.x) = r for x = or E A”. Then 0 = n,(F) 
and R = x&Y). If IT: X+ UQ’ is the canonical quotient map, then the 
assumption on the support of m assures ~(3) = xc(Z) a.s. The proof of 
Theorem 11 now follows directly from (4.1). In particular, if m = mY is U- 
invariant then mR = fi, mo = da is uniformly distributed, and m = m, x mR 
so 0 and R are independent. Conversely, if m = m8 x nzR and mB is 
uniformly distributed, then m, = da and clearly m is U-invariant. 
In the classical case, the stochastic representations have been used (cf. 
Cambanis et al. [4]) to develop the theory of elliptically contoured dis- 
tributions under minimal regularity conditions. Theorem 12 opens the way 
towards generalizations of the classical results to orthogonal transfor- 
mation groups. These problems will be treated in a later article. 
REFERENCES 
[ 11 ANDERSON, T. W. AND FANG. K. T. (1982). On the Theory of Multivariate Elliptically 
Contoured Distributions and their Applications. Technical Report No. 54, Department of 
Statistics, Stanford University. 
[2] ANDERSSON. S. (1982). Distribution of maximal invariants using proper action and 
quotient measures. Ann. Statisr. 10 955-961. 
[3] BOURBAKL N. (1963). Infkgration. Vol. XXIX, Livre VI, Chap. 7. Hermann, Paris. 
[4] CAMBANIS, S., HUANG, S., AND SIMONS, G. (1981). On the theory of elliptically con- 
toured distributions. J. Multivariate Anal. 11 368-385. 
[S] DAWID. A. P. (1978). Extendibility of spherical matrix distributions. J. Mullirrariare 
Anal. 8 559-566. 
[6] GANGOLLI, R. (1967). Positive delinite kernels on homogeneous spaces and certain 
processes related to Levy’s Brownian motion of several parameters. .4nn. hsr. H. Poin- 
cart! 3 121-225. 
[7] GROSS, K. I., HOLMAN, W. J., AND KUNZE, R. A. (1979). A new class of Bessel 
functions and applications in harmonic analysis. In Proc. SFmpos. Pure Math. Vol. 35, 
pp. 407-415. Amer. Math. Sot. Providence, R.I. 
[8] GROSS, K. I. AND KUNZE. R. A. (1976). Bessel functions and representation theory, I. 
J. Funcr. Anal. 22 73-105. 
[9] HERZ, C. S. (1955). Bessel functions of matrix argument. Ann. of Math. 61 474-523. 
[lo] JAMES, A. T. (1964). Distributions of matrix variates and latent roots derived from nor- 
mal samples. Ann. Malh. Stat&f. 35 475-501. 
[ll] PARTHASARATHY, K. R., RANGA RAO. R., AND VARADHAN, S. R. S. (1963). Probability 
distributions on locally compact abelian groups. Illinois J. Math. 7 337-369. 
138 GROSS AND RICHARDS 
[ 121 RICHAKLB, D. ST. P. ( 1986). Positive definite symmetric functions on finite dimensional 
spaces. 1. Applications of the Radon transform. J. A4ultiuariute Anal. 19 280-298. 
[ 131 RICHARIX, D. ST. P. (1984). Hyperspherical models. fractional derivatives, and 
exponential distributions on matrix spaces. Sunkhyc? 46 (Ser. A) 155-165. 
1141 RUDIN. W. (1962). Fourier Analrsis on Groups. Interscience. New York. 
[ 151 SCHOENBERG. I. J. (1938). Metric spaces and completely monotone functions, Ann. of 
Math. 39 81 t-841. 
[ 163 WEIL, A. ( 195 1). L’integration dam 1e.r groupe.~ topologiques et ses applications. 2nd ed. 
Hermann. Paris. 


Carolina Digital Repository

JOURNAL OF MULTIVARIATE ANALYSIS 21, 128-138 ( 1987) Compact Group Actions, Spherical Bessel Functions, and Invariant Random Variables* KENNETH I. GROSS’ Division of Mathematical Sciences, National Science Foundation, 1800 G Street N. W., Washington, D.C. 20550 AND DONALD ST. P. RICHARDS Department of Statistics, llniversity of North Carolina, Chapel Hill, North Carolina 27514 Communicated by G. Kallianpur The theory of compact group actions on locally compact abelian groups provides a unifying theory under which different invariance conditions studied in several contexts by a number of statisticians are subsumed as special cases. For example, Schoenberg’s characterization of radially symmetric characteristic functions on Iw” is extended to this general context and the integral representations are expressed in terms of the generalized spherical Bessel functions of Gross and Kunze. These same Bessel functions are also used to obtain a variant of the Lkvy-Khinchine formula of Parthasarathy, Ranga Rao, and Varadhan appropriate to invariant distributions. (1 1987 Academx Press. Inc 0. INTRODUCTION The theory of compact group actions on locally compact abelian groups, as developed in [8], provides a general framework for the study of invariant distributions and the unification of a variety of different kinds of AMS 1980 subject classifications: Primary 60E05, 62H10, 43A80; Secondary 62E15, 33A65. Key words and phrases: Compact groups, spherical Bessel function, Levy-Khinchine and stochastic representations, quotient measure. * Supported in part by the National Science Foundatin. Grants MCS-8101545 and MCS- 8403381. + On leave from the University of Wyoming. 128 0047-259X/87 $3.00 Copyrtght tc 1987 by Academic Press. Inc. All rights of reproduction in any lorm reserved. COMPACT GROUP ACTIONS 129 invariance considered in the literature. For example, Schoenberg’s charac- terization [ 151 of radially symmetric distributions, when considered from our viewpoint, may be regarded as classifying the positive definite functions on [w” that are invariant under the natural action of the group O(n) of orthognal n x n matrices. In this classical situation, Schoenberg showed that the characteristic function of a radially symmetric distribution can be represented by an integral of certain Bessel functions. With this in mind we generalize Schoenberg’s results from the orthogonal action on [w” to a compact group U acting on a locally compact abelian topological group .Y. Our work, therefore, gives a general context for the investigations of Dawid [IS], Anderson and Fang [l], James [IO], and other statisticians who have considered specific orthogonal actions on Euclidean spaces as they pertain to invariant distributions. In the extension of Schoenberg’s results to the compact transformation group (U, X), the kernel for realizing the characteristic function of a U- invariant distribution is the spherical Bessel function for the transformation group (U, X) that was defined and studied by Gross and Kunze [8]. This spherical Bessel function also relates to other statistical problems. For example, it allows us to adapt the work of Parthasarathy, Ranga Rao, and Varadhan [ 1 l] to prove a Levy-Khinchine formula for infinitely divisible U-invariant characteristic functions on X. This paper is organized as follows. The context for the first eight chap- ters is the general theory of a compact topological transformation group (U, X) in which the compact group U acts by automorphisms on a locally compact abelian topological group X. In particular, in Chapters 3 and 4 we study the orbit space i&Y and describe the decomposition theorem for a U-invariant measure, a result crucial to much of what follows. Chapter 5 and 6 deal with the spherical Bessel function and the generalized Schoen- berg theorem for (U, X). Our LevyyKhinchine formula is the subject of Chapters 7 and 8. The last four chapters are specialized to the case in which X is a linite- dimensional real inner product space and U is an orthogonal action on X. Chapter 9 deals with a general form of “polar coordinates” for (U, X), and Chapter 10 shows how the abstract theory of these orthogonal transfor- mation groups applies to a variety of examples including the classical con- text of Schoenberg; its generalization to matrix spaces considered by James, Richards, and others; the “rotatable” distributions of Dawid; and the invariance models of Anderson and Fang. The paper concludes with a general result in Chapters 11 and 12 on stochastic representations for U- invariant random variables. 130 GROSS AND RICHARDS 1. FOURIER ANALYSIS ON LOCALLY COMPACT ABELIAN GROUPS Throughout the paper X denotes a locally compact abelian (Hausdorff) topological group. We recall that the dual group X? consists of all con- tinuous homomorphisms of X into the multiplicative group of complex numbers of unit modulus. Then X itself becomes a locally compact abelian (Hausdorff) group relative to the compact-open topology. For /? 6 8 and .Y E X, we write (fi 1 .u) for 8(-u). Pontryagin duality identifies (X)^ with X through the formula (x I/?) = (/I 1 x) for x E X and b E X. For a finite positive Bore1 measure m on X, its Fourier transform is the function ti on 8 given by The mapping m --) GI is one-to-one. If m is a probability measure then ti is called its characteristic .function. We refer to [ 14, 161 for details. 2. COMPACT ACTIONS Throughout, U denotes a compact group acting on X by automorphisms. That is to say, (U, X) is a compact transformation group. Thus, with u(x) denoted by UX, the following properties hold: The mapping (u, x) -+ ux is continuous from U x X to X, (uIuz) x = u,(u~x) for all x in X and U, , uz in U; and x + ux is a continuous automorphism of the group X for each u E U. By duality, (U, 8) is also a compact transformation group with the action defined by (~PIx)=(Pl~-‘x) (2.1) for (u, 8, x) tz U x Bx X. 3. THE ORBIT SPACE AND INVARIANCE The orbit space for the action of U on X, denoted cr\X, consists of all orbits Ux = {ax: u E U}. Equipped with the quotient topology the orbit space is a locally compact Hausdortf space. The canonical quotient map rtc: x -+ Ux is open and continuous from X to V/X. A function f on X is called U-invariant if f(ux) =f(x) for all (u, x) E U x X. Equivalently, such an f is constant on orbits. This means that a U-invariant function f can be regarded as a function f on cr\X, where J(UL=J: (3.1) COMPACTGROUPACTIONS 131 In order to construct U-invariant functions, let f be any continuous function f on X. Define (3.2) where du is normalized Haar measure on U. By the right invariance of Haar measure, f, is U-invariant. By standard measure theory, formula (3.2) is defined almost surely in X for f~ L’(X, m). Similarly, a measure m on X is U-invariant if m(z&) = m(B) for all 2.4 E U and Bore1 sets B in X. The following theorem shows that there is a one-to- one correspondence between invariant measures m on X and measures G on v\X. See [3, Sect. 2; 2, Sect. 3; or 8, Sect. 21. 4. THEOREM Let m be a U-invariant measure on X. Then there exists a unique measure 6 on U/X such that for all f e L’(X, m). In other words, ~*f(x)dm(x)=~~,.IS,(n(x))~~(x(.r)). (4.2) The measure fi on v\X is called the quotient measure corresponding to the U-invariant measure m. This “decomposition” of an invariant measure is crucial to later results. 5. THE SPHERICAL BESSEL FUNCTION AND SCHOENBERG'S THEOREM The spherical Bessel function for the transformation group (17, X) is the function JO on xx X defined by JcdB, x) = jL, (B I ux) du (5.1) for (j?, x) E 8x X. Note that JO is U-invariant in each variable; that is, JcdB, x) = Jo(uB, x) = JdB, ux) (5.2) for all (u,~,x)~UxRxX. 683’?1’1-Y 132 GROSS AND RICHARDS The spherical Bessel functions were defined in [S] in connection with the Fourier analysis of (U, X). To briefly indicate their significance, note that for a U-invariant finite measure m on X, h(B) = 1 Jo@, xl dm(-xl (5.3) x for all /3 E 2. Indeed, since m is U-invariant, so is k; for if u E U and /I E 2 then rh(u/3)=S (uj?Ix)dm(x)=j (plu-‘x)dm(x) 2 x Therefore. = tP I ux) du Wx) = jxJo@, x) dm(x). The following theorem generalizes the decomposition theorem for IL!“, first proved by Schoenberg [ 141. 6. THEOREM Let m be a U-invariant probability measure on X and let fi be the corresponding quotient measure on v\X. Then S? is a probability measure, and +4/n = j JdB, x) d@44x)) U\,l for /I E 2. Proof: We apply Theorem 4 to (5.3). Thus, for U-invariant m, W?) = Ix JdP, xl dm(x) = s JdP, x) dfi(4x)) U\X COMPACTGROUPACTIONS 133 by (5.2) and the fact that du has mass 1. All that remains is to verify, by the substitution f= 1 in Theorem 4, that fi is a probability measure if m is a probability measure. 7. THE SPHERICAL BESSEL FUNCTION AND THE LEVY-KHINCHINE FORMULA We relate the spherical Bessel function for (U, X) to infinitely divisible probability measures. Our version of the Levy-Khinchine representation relies upon the work of Parthasarathy, Ranga Rao, and Varadhan [ 111, but in part is motivated by the variant of their theorem due to Gangolli [6]. Let X be separable and metric and p a probability measure on X. Following [ 111, we say p is idempotent if p * p = p * 6, for some x in X, where * denotes convolution and 6, is the Dirac measure at x. The measure pI is said to be a factor of p if p = p1 * pLz for some measure pz. Also, p is called infinite/y divisible if for each positive integer n, there exist x, in X and a measure 1, such that p = A; * 6,“, where 1; = I, * I, * . * 1, (n convolutions). The proof of the following Levy-Khinchine theorem is derived from Theorem 7.1 of [ 111 by applying the “radializing” formula, (3.2), above. 8. THEOREM Let X be a locally compact abelian separable metric group, U be as before, and m be an infinitely divisible, U-invariant probability measure on X. If m has no idempotent factors and m is real-valued, then m has the representation The measure F is a-finite on X; it has finite mass outside every neighborhood of the identity in X; and s Cl -Jo@, x)1 dF(x) < ~0 x for all BEX. Also, D(p) is a non-negative, continuous U-invariant function satisfying @(PI + Bd + WI - P2) = 2C@v1) + @(B*)l 134 GROSS AND RICHARDS 9. ORTHOGONAL ACTIONS AND POLAR COORDINATES Suppose now that X is a finite dimensional real vector space with an inner product and that the elements of U act orthogonally on X; i.e., ljuxll = Ilxll for all (u, X) E U x X, where llj is the norm derived from the inner product. In this case one calls (U, X) an orthogonal transformation group. Throughout the remainder of the paper we assume (U, X) is an orthogonal transformation group. By polar coordinates for (U, X) we mean a decomposition C x C for a dense open subset x’ of X whose complement has Lebesgue measure zero. More specifically, we suppose that Z= UJ’U, for a fixed closed subgroup UO of U, C is an open subset of the subspace X,, of points in X fixed by UO, for each rE C the orbit Ur is equivalent to C, and the mapping (uU,, r) -+ ur is a diffeomorphism of Z x C onto x’. The points in X’ are called regular. In the presence of polar coordinates, the orbit space (except for a null set) can be identified with C, and the spherical Bessel function is deter- mined more explicitly on C. The following examples illustrate this phenomenon. 10. EXAMPLES (A) Let X= 08” with the usual inner product (.I.) and U= O(n), the group of n x n orthogonal matrices. U acts on X by left matrix mul- tiplication, x + UX, u E U, x E X. The dual group J? may be identified with R” via the inner product, since every element of 2 is of the form exp( i < 0, .) ), 0 E KY’. Here, we have the usual polar coordinates: X’= {XE R”: x #O}; .E is the unit sphere, realized as the set UC,, where e0 = (1, O,..., 0)’ and U,=(uEU:uo,=o,}rO(n-1); and C= { roO E R”: r > 0} E (0, co). Thus, for x # 0 in X, z(x) is identified with the number r = lixlj. If /I, XE R”, then where Q,(t) = 2’r(n/2) t~‘.l,(t), t ~0. Here, v = (n-2)/2, and .I, is the classical Bessel function of the first kind of order v. Putting these facts COMPACT GROUP ACTIONS 135 together, we recover Schoenberg’s classical result from Theorem 6: If m is an O(n)-invariant probability measure on R”, then G(x) = $( l]xll), where and ti is a probability measure on [0, co). Here, we call the reader’s attention to the fact that the function Jo, originally defined on R” x Iw” is ultimately determined by a function on R. (B) For n>k, let X= lRnxk, the space of n x k real matrices, with inner product (x, y) = tr(x’y), where x’ denotes the transpose of x, and U = O(n). The group U acts “one-sidely” on X by left matrix mul- tiplication, x --t UX, u E U, x E X. 8 may be identified with X since the homomorphisms in f are of the form exp(i trace(B’x)). Here, Y= {x E X: x has maximal rank, k), C = C,,, is the Stiefel manifold {(r E [w” x k: C+C = 1, > which is realized as UCY,,, where CT~ is the n x k matrix with (o,),~= 1 for all j = l,..., k and all other entries are zero; U, = {u~U:uo,=o,}~O(n-k); and c={a,r~X: r is a kxk (symmetric) positive definite matrix ). Thus, for x E aB”” k such that r = x’x is invertible, n(x) is identified with the positive definite k x k matrix r’/‘. If /I, x E R” x k, then Joa -xl = co,,, exp(i trace(B’ux)) ~2.4 = constant x A ,,(~x’xfi’j?), where A _ ,,J.) is the Bessel function of matrix argument defined by Herz [9]; see also [S]. The corresponding generalization of Schoenberg’s theorem now reads: If m is an O(n)-invariant probability measure on R” x k, then G(x) =&x’x), where for any t E {t 30}, the cone of symmetric positive definite k x k matrices, 4(t) = constant x s A lj2(tr) dfi(r), r>O and fi is a probability measure on {r 2 0). We remark that Example A is the special case k = 1. Observe, again, the phenomenon that the function J,, originally defined for n x k matrices is determined by a function on the lower dimensional space of k x k matrices. We note in passing that random matrices having distributions of the type treated in this example have been studied in the work of Richards [ 131 and others. 136 GROSS AND RICHARDS (C) Let x= 1w;y”m”, the space of n x n real symmetric matrices. The group U = O(n) acts “two-sidedly” on X via ,Y + uxu ‘, ME U, XE X. As in the previous example, it may be shown that X can be identified with X. In addition, v\X may be identified with the set A,, of n x n real diagonal matrices. For /?, .Y E X, JO(P? -u) = j(),,l, exp( i trace(pusu ‘,)du=,F,(iP,x), where ,-,F, is the hypergeometric function of two matrix arguments defined by James [lo]. It is straightforward to write down the generalized Schoen- berg theorem for this example. Random matrices having two-sided invariant distributions have been studied by Dawid [5] who calls such distributions rotatable. (D) Let X= X= [Wnxk and U = O(n) x .. . x O(n), the direct product of k copies of O(n). Let x, ,..., xk be the columns of x E X and u = (u, ,..., u,) be a typical element of U, where ui E O(n) for all i = I,..., k. The group U acts on X via x = [x, ,..., sk] -+ [u,x,, u2x2 ,..., ukxk]. This is just k copies of the Euclidean example A. If B = c/l, ,..., pk] E 2, and x = [.x, ,..., xk] E X, then where a,,(.) is defined in Example A. From this, we immediately derive a Schoenberg-type theorem obtained previously by Anderson and Fang Cl]. We also note that examples analogous to those above can be constructed for matrices with entries in any real finite-dimensional division algebra [F. When IF = @, the spherical Bessel functions are related to the complex generalized hypergeometric functions of James [lo]; see also [7, 81. 11. STOCHASTIC REPRESENTATIONS In the classical situation, Schoenberg’s theorem is sometimes restated in terms of stochastic representations. We shall now extend this concept to the group setting using polar coordinates for the orthogonal transfor- mation group (U, X). COMPACT GROUP ACTIONS 137 12. THEOREM Let 2 be an X-valued random variable with associated probability measure m. Further, assume that m is supported on the set x’ of regular elements of X, where 3 = OR is the unique decomposition of 3 through polar coor- dinates. Then m is U-invariant if and only if0 and R are independent, and 0 is uniformly distributed with respect to the normalized U-invariant measure on z= u/u,,. Here, we write nZ(x) = CJ and nC(.x) = r for x = or E A”. Then 0 = n,(F) and R = x&Y). If IT: X+ UQ’ is the canonical quotient map, then the assumption on the support of m assures ~(3) = xc(Z) a.s. The proof of Theorem 11 now follows directly from (4.1). In particular, if m = mY is U- invariant then mR = fi, mo = da is uniformly distributed, and m = m, x mR so 0 and R are independent. Conversely, if m = m8 x nzR and mB is uniformly distributed, then m, = da and clearly m is U-invariant. In the classical case, the stochastic representations have been used (cf. Cambanis et al. [4]) to develop the theory of elliptically contoured dis- tributions under minimal regularity conditions. Theorem 12 opens the way towards generalizations of the classical results to orthogonal transfor- mation groups. These problems will be treated in a later article. REFERENCES [ 11 ANDERSON, T. W. AND FANG. K. T. (1982). On the Theory of Multivariate Elliptically Contoured Distributions and their Applications. Technical Report No. 54, Department of Statistics, Stanford University. [2] ANDERSSON. S. (1982). Distribution of maximal invariants using proper action and quotient measures. Ann. Statisr. 10 955-961. [3] BOURBAKL N. (1963). Infkgration. Vol. XXIX, Livre VI, Chap. 7. Hermann, Paris. [4] CAMBANIS, S., HUANG, S., AND SIMONS, G. (1981). On the theory of elliptically con- toured distributions. J. Multivariate Anal. 11 368-385. [S] DAWID. A. P. (1978). Extendibility of spherical matrix distributions. J. Mullirrariare Anal. 8 559-566. [6] GANGOLLI, R. (1967). Positive delinite kernels on homogeneous spaces and certain processes related to Levy’s Brownian motion of several parameters. .4nn. hsr. H. Poin- cart! 3 121-225. [7] GROSS, K. I., HOLMAN, W. J., AND KUNZE, R. A. (1979). A new class of Bessel functions and applications in harmonic analysis. In Proc. SFmpos. Pure Math. Vol. 35, pp. 407-415. Amer. Math. Sot. Providence, R.I. [8] GROSS, K. I. AND KUNZE. R. A. (1976). Bessel functions and representation theory, I. J. Funcr. Anal. 22 73-105. [9] HERZ, C. S. (1955). Bessel functions of matrix argument. Ann. of Math. 61 474-523. [lo] JAMES, A. T. (1964). Distributions of matrix variates and latent roots derived from nor- mal samples. Ann. Malh. Stat&f. 35 475-501. [ll] PARTHASARATHY, K. R., RANGA RAO. R., AND VARADHAN, S. R. S. (1963). Probability distributions on locally compact abelian groups. Illinois J. Math. 7 337-369. 138 GROSS AND RICHARDS [ 121 RICHAKLB, D. ST. P. ( 1986). Positive definite symmetric functions on finite dimensional spaces. 1. Applications of the Radon transform. J. A4ultiuariute Anal. 19 280-298. [ 131 RICHARIX, D. ST. P. (1984). Hyperspherical models. fractional derivatives, and exponential distributions on matrix spaces. Sunkhyc? 46 (Ser. A) 155-165. 1141 RUDIN. W. (1962). Fourier Analrsis on Groups. Interscience. New York. [ 151 SCHOENBERG. I. J. (1938). Metric spaces and completely monotone functions, Ann. of Math. 39 81 t-841. [ 163 WEIL, A. ( 195 1). L’integration dam 1e.r groupe.~ topologiques et ses applications. 2nd ed. Hermann. Paris. 

Compact group actions, spherical bessel functions, and invariant random variables

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Compact group actions, spherical bessel functions, and invariant random variables

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