The Lukacs–Olkin–Rubin Theorem on Symmetric Cones Without Invariance of the “Quotient”
Bartosz Koodziejek
Mathematics Subject Classification
We prove the LukacsOlkinRubin theorem without invariance of the distribution of the quotient, which was the key assumption in the original proof of (OlkinRubin in Ann Math Stat 33:12721280, 1962). Instead, we assume existence of strictly positive continuous densities of respective random variables. We consider the (cone variate) quotient for any division algorithm satisfying some natural conditions. For that purpose, a new proof of the OlkinBaker functional equation on symmetric cones is given. The Lukacs [17] theorem is one of the most celebrated characterizations of probability distributions. It states that if X and Y are independent, positive, nondegenerate random variables such that their sum and quotient are also independent, then X and Y have gamma distributions with the same scale parameter. This theorem has many generalizations. The most important in the multivariate setting was given by Olkin and Rubin [21] and Casalis and Letac [7], where the authors extended characterization to matrix and symmetric cones variate distributions, respectively. There is no unique way of defining the quotient of elements of the cone

of positive definite symmetric matrices +, and in these papers, the authors have
considered very general form U = g( X + Y ) X gT ( X + Y ), where g is the socalled
division algorithm, that is, g(a) a gT (a) = I for any a +, where I is the identity
matrix and g(a) is invertible for any a + (later on, abusing notation, we will write
g(x)y = g(x) y gT (x), that is, in this case, g(x) denotes the linear operator acting
on +). The drawback of their extension was the additional strong assumption of
invariance of the distribution of U under a group of automorphisms. This result was
generalized to homogeneous cones in Boutouria et al. [6].
There were successful attempts in replacing the invariance of the quotient
assumption with the existence of regular densities of random variables X and Y . Bobecka and
Wesoowski [2] assuming existence of strictly positive, twice differentiable densities
proved a characterization of Wishart distribution on the cone + for division
algorithm g1(a) = a1/2, where a1/2 denotes the unique positive definite symmetric root
of a +. These results were generalized to all nonoctonion symmetric cones of
rank >2 and to the Lorentz cone for strictly positive and continuous densities by
Koodziejek [12, 13].
Exploiting the same approach, with the same technical assumptions on densities
as in Bobecka and Wesoowski [2], it was proven by Hassairi et al. [11] that the
independence of X + Y and the quotient defined through the Cholesky decomposition,
i.e., g2(a) = Ta1, where Ta is a lower triangular matrix such that a = Ta TaT +,
characterizes a wider family of distributions called Riesz (or sometimes called
RieszWishart). This fact shows that the invariance property assumed in Olkin and Rubin
[21] and Casalis and Letac [7] is not of technical nature only. Analogous results for
homogeneous cones were obtained by Boutouria [4, 5].
In this paper, we deal with the density version of LukacsOlkinRubin theorem
on symmetric cones for division algorithm satisfying some natural properties. We
assume that the densities of X and Y are strictly positive and continuous. We consider
quotient U for an arbitrary, fixed division algorithm g as in the original paper of Olkin
and Rubin [21], additionally satisfying some natural conditions. In the known cases
(g = g1 and g = g2), this improves the results obtained in Bobecka and Wesoowski,
Hassairi et al. and Koodziejek [2, 11, 13]. In general case, the densities of X and Y
are given in terms of, socalled, wmultiplicative Cauchy functions, that is functions
satisfying
f (x) f (w(I )y) = f (w(x)y) , (x, y)
where w(x)y = w(x) y wT (x) (i.e., g(x) = w(x)1 is a division algorithm).
Consistently, we will call w a multiplication algorithm. Such functions were recently
considered in Koodziejek [14].
Unfortunately, we cannot answer the question whether there exists division (or
equivalently multiplication) algorithm resulting in characterizing other distribution
than Riesz or Wishart. Moreover, the simultaneous removal of the assumptions of the
invariance of the quotient and the existence of densities remains a challenge.
This paper is organized as follows. We start in the next section with basic definitions
and theorems regarding analysis on symmetric cones. The statement and proof of the
main result are given in Sect. 4. Section 3 is devoted to consideration of wlogarithmic
2 Preliminaries
Cauchy functions and the OlkinBaker functional equation. In that section, we offer
much shorter, simpler and covering more general cones proof of the OlkinBaker
functional equation than given in Bobecka and Wesoowski, Hassairi et al. and Koodziejek
[2,11,13].
In this section, we give a short introduction to the theory of symmetric cones. For
further details, we refer to Faraut and Kornyi [8].
A Euclidean Jordan algebra is a Euclidean space E (endowed with scalar product
denoted x, y ) equipped with a bilinear mapping (product)
(x, y) xy E
and a neutral element e in E such that for all x, y, z in E:
For x E let L(x) : E E be linear map defined by
L(x)y = xy,
P(x) = 2L2(x) L x2 .
The map P : E E nd(E) is called the quadratic representation of E.
An element x is said to be invertible if there exists an element y in E such that
L(x)y = e. Then, y is called the inverse of x and is denoted by y = x1. Note that
the inverse of x is unique. It can be shown that x is invertible if and only if P(x) is
invertible, and in this case, (P(x))1 = P x1 .
Euclidean Jordan algebra E is said to be simple if it is not a Cartesian product
of two Euclidean Jordan algebras of positive dimensions. Up to linear isomorphism,
there are only five kinds of Euclidean simple Jordan algebras. Let K denotes either
the real numbers R, the complex ones C, quaternions H or the octonions O and write
Sr (K) for the space of r r Hermitian matrices with entries valued in K, endowed
with the Euclidean structure x, y = Trace (x y ) and with the Jordan product
where x y denotes the ordinary product of matrices and y is the conjugate of y. Then,
Sr (R), r 1, Sr (C), r 2, Sr (H), r 2, and the exceptional S3(O) are the first four
kinds of Euclidean simple Jordan algebras. Note that in this case
xy = 21 (x y + y x),
P(y)x = y x y.
(x0, x1, . . . , xn) (y0, y1, . . . , yn) =
xi yi , x0 y1 + y0x1, . . . , x0 yn + y0xn . (3)
To each Euclidean simple Jordan algebra, one can attach the set of Jordan squares
i=0
=
x2 : x E .
The interior is a symmetric cone. Moreover, is irreducible, i.e., it is not the
Cartesian product of two convex cones. One can prove that an open convex cone is
symmetric and irreducible if and only if it is the cone of some Euclidean simple
Jordan algebra. Each simple Jordan algebra corresponds to a symmetric cone; hence,
there exist up to linear isomorphism also only five kinds of symmetric cones. The cone
corresponding to the Euclidean Jordan algebra Rn+1 equipped with Jordan product
(3) is called the Lorentz cone.
We denote by G(E) the subgroup of the linear group G L(E) of linear
automorphisms which preserves , and we denote by G the connected component of G(E)
containing the identity. Recall that if E = Sr (R) and G L(r, R) is the group of
invertible r r matrices, elements of G(E) are the maps g : E E such that there exists
a G L(r, R) with
g(x) = a x aT .
K = {k G : ke = e}.
A multiplication algorithm is a map G : x w(x) such that w(x)e = x
for all x . This concept is consistent with, socalled, division algorithm g, which
was introduced by Olkin and Rubin [21] and Casalis and Letac [7], that is a mapping
x g(x) G such that g(x)x = e for any x . If w is a multiplication
algorithm, then g = w1 (that is, g(x)w(x) = w(x)g(x) = I d for any x ) is
a division algorithm and vice versa; if g is a division algorithm, then w = g1 is a
multiplication algorithm. One of two important examples of multiplication algorithms
is the map w1(x) = P x1/2 .
We will now introduce a very useful decomposition in E, called spectral
decomposition. An element c E is said to be a idempotent if cc = c = 0. Idempotents
a and b are orthogonal if ab = 0. Idempotent c is primitive if c is not a sum of two
nonnull idempotents. A complete system of primitive orthogonal idempotents is a set
(c1, . . . , cr ) such that
ci = e and ci c j = i j ci for 1 i j r.
The size r of such system is a constant called the rank of E. Any element x of a
Euclidean simple Jordan algebra can be written as x = ri=1 i ci for some complete
system of primitive orthogonal idempotents (c1, . . . , cr ). The real numbers i , i =
1, . . . , r are the eigenvalues of x. One can then define trace and determinant of x by,
respectively, tr x = ri=1 i and det x = ri=1 i . An element x E belongs to if
and only if all its eigenvalues are strictly positive.
The rank r and dim of irreducible symmetric cone are connected through relation
= r +
where d is an integer called the Peirce constant.
If c is a primitive idempotent of E, the only possible eigenvalues of L(c) are 0, 21
and 1. We denote by E(c, 0), E(c, 21 ) and E(c, 1) the corresponding eigenspaces. The
decomposition
E = E(c, 0) E(c, 21 ) E(c, 1)
is called the Peirce decomposition of E with respect to c. Note that P(c) is the
orthogonal projection of E onto E(c, 1).
Fix a complete system of orthogonal idempotents (ci )ri=1. Then for any i, j
{1, 2, . . . , r }, we write
1
E c j , 2
if i = j.
It can be proved (see Faraut and Kornyi [8, Theorem IV.2.1]) that
E =
i j
Ei j Ei j Eii + Ei j ,
Ei j E jk Eik , if i = k,
Ei j Ekl = {0}, if {i, j } {k, l} = .
x2 = 21 x 2(ci + c j ),
xy 2 = 18 x 2 y 2.
The dimension of Ei j is the Peirce constant d for any i = j . When E is
Sr (K), if (e1, . . . , er ) is an orthonormal basis of Rr , then Eii = Rei eiT and Ei j
= K ei e Tj + e j eiT for i < j and d is equal to di mRK.
For 1 k r , let Pk be the orthogonal projection onto E(k) = E(c1 + . . . + ck , 1),
det(k) the determinant in the subalgebra E(k), and, for x , k (x) = det(k)( Pk (x)).
Then, k is called the principal minor of order k with respect to the Jordan frame
(ck )rk=1. Note that r (x) = det x. For s = (s1, . . . , sr ) Rr and x , we write
s (x) =
1(x)s1s2 2(x)s2s3 . . . r (x)sr .
s is called a generalized power function. If x =
2s2 . . . rsr .
We will now introduce some basic facts about triangular group. For x and y in ,
let x y denote the endomorphism of E defined by
x y = L(xy) + L(x)L(y) L(y)L(x).
If c is an idempotent and z E(c, 21 ), we define the Frobenius transformation c(z)
in G by
x = c1 (z(1))c2 (z(2)) . . . cr1 (z(r1))
k=1
Mapping w2 : T , x w2(x) = tx realizes a multiplication algorithm.
For E = Sr (R), we have = +. Let us define for 1 i, j r matrix i j
= (kl )1k,lr such that i j = 1 and all other entries are equal 0. Then for Jordan
Given a Jordan frame (ci )ri=1, the subgroup of G,
i=1
k= j+1
E jk
is called the triangular group corresponding to the Jordan frame (ci )ri=1. For any x
in , there exists a unique tx in T such that x = txe, that is, there exist (see Faraut
and Kornyi [8, Theorem IV.3.5]) elements z( j) rk= j+1 E jk , 1 j r 1 and
positive numbers 1, . . . , r such that
frame (ci )ri=1, where ck = kk , k = 1, . . . , r , we have z jk = ( jk + k j ) E jk
oraz z jk 2 = 2, 1 j, k r , j = k. if z(i) rj=i+1 Ei j , i = 1, . . . , r 1, then
there exists (i) = (i+1, . . . , r ) Rri such that z(i) = rj=i+1 j zi j . Then, the
Frobenius transformation reads
ci (z(i))x = Fi ((i)) x Fi ((i))T ,
j=i+1
It can be shown (Faraut and Kornyi [8, Proposition VI.3.10]) that for each t T ,
x and s Rr ,
and for any z E(ci , 21 ), i = 1, . . . , r ,
s (t x) =
if only s and T are associated with the same Jordan frame (ci )ri=1.
We will now introduce some necessary basics regarding certain probability
distribution on symmetric cones. Absolutely continuous Riesz distribution Rs,a on
is defined for any a and s = (s1, . . . , sr ) Rr such that si > (i 1)d/2,
i = 1, . . . , r , though its density
Rs,a(dx) =
sdim / r (x)e a,x I (x) dx, x ,
where s is the generalized power function with respect to a Jordan frame (ci )ri=1
and is the Gamma function of the symmetric cone . It can be shown that (s)
= (2 )(dim r)/2 rj=1 ( s j ( j 1) d2 ) (see Faraut and Kornyi [8, VII.1.1.]). Riesz
distribution was introduced in Hassairi and Lajmi [10].
Absolutely continuous Wishart distribution p,a on is a special case of Riesz
distribution for s1 = . . . = sr = p. If a and p > dim / r 1 it has density
(det x) pdim / r e a,x I (x) dx, x ,
where ( p) := ( p, . . . , p). Wishart distribution is a generalization of gamma
distribution (case r = 1).
In generality, Riesz and Wishart distributions does not always have densities, but
due to the assumption of existence of densities in Theorem 4.2, we are not interested
in other cases.
3 Functional Equations
3.1 Logarithmic Cauchy Functions
As will be seen, the densities of respective random variables will be given in terms of
wlogarithmic Cauchy functions, i.e., functions f : R that satisfy the following
functional equation
f (x) + f (w(e)y) = f (w(x)y), (x, y)
where w is a multiplication algorithm. If f is wlogarithmic, then e f is called
wmultiplicative. In the following section, we will give the form of wlogarithmic Cauchy
functions for two basic multiplication algorithms, one connected with the quadratic
representation
w1(x) = P(x1/2),
w2(x) = tx T .
and the other related to a triangular group T ,
Such functions were recently considered without any regularity assumptions in
Koodziejek [14].
It should be stressed that there exists infinite number of multiplication algorithms. If
w is a multiplication algorithm, then trivial extensions are given by w(k)(x) = w(x)k,
where k K is fixed (Remark 4.3 explains why this extension is trivial when it comes
to multiplicative functions). One may consider also multiplication algorithms of the
form P(x)tx12 , which interpolate between the two main examples: w1 (which is
= 1/2) and w2 (which is = 0). In general, any multiplication algorithm may be
written in the form w(x ) = P(x1/2)kx , where x kx K .
Functional equation (9) for w1 was already considered by Bobecka and Wesoowski
[3] for differentiable functions and by Molnr [19] for continuous functions of real
or complex Hermitian positive definite matrices of rank >2. Without any regularity
assumptions, it was solved on the Lorentz cone by Wesoowski [23].
Case of w2(x) = tx T for a triangular group T , perhaps a bit surprisingly, leads
to a different solution. It was indirectly solved for differentiable functions by Hassairi
et al. [11, Proof of Theorem 3.3].
By Faraut and Kornyi [8, Proposition III.4.3], for any g in the group G,
det(gx) = (Det g)r/ dim
where Det denotes the determinant in the space of endomorphisms on . Inserting a
multiplication algorithm g = w(y), y , and x = e, we obtain
Det (w(y)) = (det y)dim / r
det(w(y)x) = det y det x
for any x, y . This means that f (x) = H (det x), where H is generalized
logarithmic function, i.e., H (ab) = H (a) + H (b) for a, b > 0, is always a solution to (9),
regardless of the choice of multiplication algorithm w. If a wlogarithmic functions f
is additionally K invariant ( f (x) = f (kx) for any k K ), then H (det x) is the only
possible solution (Theorem 3.4).
In Koodziejek [14], the following theorems have been proved. They will be useful
in the proof of the main theorems in this paper.
Theorem 3.1 (w1logarithmic Cauchy functional equation) Let f :
function such that
f (x) + f (y) = f P x1/2 y , (x, y)
Then, there exists a logarithmic function H such that for any x
Theorem 3.2 (w2logarithmic Cauchy functional equation) Let f :
function satisfying
for any x and y in the cone of rank r , ty T , where T is the triangular group
with respect to the Jordan frame (ci )ri=1. Then, there exist generalized logarithmic
functions H1, . . . , Hr such that for any x ,
f (x) = H (det x).
f (x) + f (y) = f (tyx)
f (x) =
k=1
k is the principal minor of order k with respect to (ci )ri=1.
If we assume in Theorem 3.2 that f is additionally measurable, then functions Hk are
measurable. This implies that there exist constants sk R such that Hk () = sk log
and
k=1
f (x) =
sk log( k (x)) = log
k=1
Thus, we obtain the following
Remark 3.3 If we impose on f in Theorem 3.2 some mild conditions (e.g.,
measurability), then there exists s Rr such that for any x ,
f (x) = log s (x).
f (x) = H (det x).
Theorem 3.4 Let f : R be a function satisfying (9). Assume additionally that
f is K invariant, i.e., f (kx) = f (x) for any k K and x . Then, there exists a
logarithmic function H such that for any x ,
Lemma 3.5 (wlogarithmic Pexider functional equation) Assume that a, b, c defined
on the cone satisfy following functional equation
a(x) + b(y) = c(w(x)y), (x, y)
Then, there exist wlogarithmic function f and real constants a0, b0 such that for any
x ,
a(x) = f (x) + a0,
b(x) = f (w(e)x) + b0,
c(x) = f (x) + a0 + b0.
3.2 The OlkinBaker Functional Equation In the following section, we deal with the OlkinBaker functional equation on irreducible symmetric cones, which is related to the Lukacs independence condition (see proof of the Theorem 4.2).
Henceforth, we will assume that multiplication algorithm w additionally is
homogeneous of degree 1, that is, w(sx) = sw(x) for any s > 0 and x . It is easy to
create a multiplication algorithm without this property, for example:
w(x) =
w1(x), if det x > 1,
w2(x), if det x 1.
The problem of solving
f (x )g(y) = p(x + y)q(x /y), (x , y) (0, )2
for unknown positive functions f , g, p and q was first posed in Olkin [20]. Note
that in onedimensional case, it does not matter whether one considers q(x /y) or
q(x /(x + y)) on the right hand side of (13). Its general solution was given in Baker
[1] and later analyzed in Lajk [15] using a different approach. Recently, in Mszros
[18] and Lajk and Mszros [16], Eq. (13) was solved assuming that it is satisfied
almost everywhere on (0, )2 for measurable functions which are nonnegative on its
domain or positive on some sets of positive Lebesgue measure, respectively. Finally,
a new derivation of solution to (13), when the equation holds almost everywhere on
(0, )2 and no regularity assumptions on unknown positive functions are imposed,
was given in Ger et al. [9]. The following theorem is concerned with an adaptation of
(13), after taking logarithm, to the symmetric cone case.
Theorem 3.6 (OlkinBaker functional equation on symmetric cones) Let a, b, c and
d be real continuous functions on an irreducible symmetric cone of rank r . Assume
a(x) + b(y) = c(x + y) + d (g (x + y) x) ,
(x, y)
where g1 = w is a homogeneous of degree 1 multiplication algorithm. Then, there
exist constants Ci R, i = 1, . . . , 4, E such that for any x and u D
= {x : e x },
a(x) =
b(x) =
c(x) = , x + e(x) + f (x) + C3,
d(u) = e(w(e)u) + f (e w(e)u) + C4,
where e and f are continuous wlogarithmic Cauchy functions and C1 + C2 =
C3 + C4.
We will need following simple lemma. For the elementary proof, we refer to
Koodziejek [13, Lemma 3.2].
Lemma 3.7 (Additive Pexider functional equation on symmetric cones) Let a, b and
c be measurable functions on a symmetric cone satisfying
a(x) + b(y) = c(x + y),
(x, y)
Then, there exist constants , R and E such that for all x
Now, we can come back and give a new proof the OlkinBaker functional equation.
Proof of Theorem 3.6 In the first part of the proof, we adapt the argument given in
Ger et al. [9], where the analogous result on (0, ) was analyzed, to the symmetric
cone setting.
For any s > 0 and (x, y) 2, we get
a(sx) + b(sy) = c(s(x + y)) + d (g(sx + sy)sx) .
Since w is homogeneous of degree 1, we have g(sx) = 1s g(x) and so g(sx + sy)sx
= g(x + y)x for any s > 0. Subtracting now (14) from (17) for any s > 0, we arrive
at the additive Pexider equation on symmetric cone ,
as (x) + bs (y) = cs (x + y),
(x, y)
where as , bs and cs are functions defined by as (x) := a(sx) a(x), bs (x) := b(sx)
b(x) and cs (x) := c(sx) c(x).
Due to continuity of a, b and c and Lemma 3.7, it follows that for any s > 0, there
exist constants (s) E, (s) R and (s) R such that for any x ,
By the definition of as and the above observation, it follows that for any (s, t )
(0, )2 and z
ast (z) = at (sz) + as (z).
Since (18) holds for any z , we see that (st ) = (s)+(t ) for all (s, t ) (0, )2.
That is (s) = k1 log s for s (0, ), where k1 is a real constant.
On the other hand
(st ), z = (s), z + (t ), sz = (t ), z + (s), t z
since one can interchange s and t on the lefthand side. Putting s = 2 and denoting
= (2), we obtain
, z (t 1)
a (x) = a(x)
a (sx) = a (x) + k1 log s
b(x) = b(x)
c(x) = c(x)
for s > 0 and x .
Analogous considerations for function bs gives existence of constant k2 such that
b(sx) = b(x) + k2 log s, where
hence c(sx) = c(x) + (k1 + k2) log s and
for any s > 0 and x .
Functions a , b, c and d satisfy original OlkinBaker functional equation:
a (x) + b(y) = c(x + y) + d (g (x + y) x) , (x, y)
Taking x = y = v
in (22), we arrive at
a (v) + b(v) = c(2v) + d( 21 e) = c(v) + (k1 + k2) log 2 + d( 21 e).
for t > 0 and z
. It then follows that for all s (0, ) and z
as (z) = a(sz) a(z) =
, z (s 1) + k1 log s.
Let us define function a by formula
From (20), we get
Insert x = w(v)u and y = w(v)(eu) into (22) for 0 < < 1 and (u, v) (D, ) .
Using (21), we obtain
Let us observe, that due to continuity of b on and lim0 {w(v)(e u)}
= w(v)e = v (convergence in the norm generated by scalar product , ),
limit as 0 of the lefthand side of the above equality exists. Hence, the limit of
the righthand side also exists and
a (w(v)u) + b(v) = c(v) + lim0 {d(u) k1 log } , (u, v) (D, ).
Subtracting (24) from (23), we have
a (w(v)u) = a (v) + g(u)
for u D, v , where g(u) = lim0 {d(u) k1 log }(k1 +k2) log 2d( 21 e).
Due to the property (21), equation (25) holds for any u , so we arrive at the
wlogarithmic Pexider equation. Lemma 3.5 implies that there exists wlogarithmic
function e such that
a (x) = e(x) + C1
for any x and a constant C1 R. Function e is continuous, because a is
continuous. Coming back to the definition of a , we obtain
a(x) =
, x + e(x) + C1, x .
Analogously for function b, considering equation (22) for x = w(v)(e u) and
y = w(v)u after passing to the limit as 0, we show that there exists continuous
wlogarithmic function f such that
b(x) =
, x + f (x) + C2, x
for a constant C2 R. The form of c follows from (23). Taking x = w(e)u and
y = e w(e)u in (22) for u D, we obtain the form of d.
4 The LukacsOlkinRubin Theorem Without Invariance of The Quotient
In the following section, we prove the density version of LukacsOlkinRubin theorem
for any multiplication algorithm w satisfying
(i) w(sx) = sw(x) for s > 0 and x
(ii) differentiability of mapping
We assume (ii) to ensure that Jacobian of the considered transformation exists. We start
with the direct result, showing that the considered measures have desired property. The
converse result is given in Theorem 4.2. For every generalized multiplication w, the
family of these wWishart measures (as defined in (26)) contains the Wishart laws. For
w = w1, there are no other distributions, while the w2Wishart measures consist of the
Riesz distributions. It is an open question whether there is a generalized multiplication
w that leads to other probability measures in this family.
Theorem 4.1 Let w be a multiplication algorithm satisfying condition (ii) and define
g = w1. Suppose that X and Y are independent random variables with densities
given by
f X (x) = C X e(x) exp , x I (x),
fY (x) = CY f (x) exp , x I (x),
where e and f are wmultiplicative functions, E and E is the Euclidean Jordan
algebra associated with the irreducible symmetric cone .
Then, vector (U, V ) = (g(X + Y )X, X + Y ) have independent components.
Note that if w(x) = w1(x) = P(x1/2), then there exist positive constants X and
Y such that e(x) = (det x)X dim / r and f (x) = (det x)Y dim / r . In this case
=: a and (X, Y ) X ,a Y ,a. Similarly, if w(x) = w2(x) = tx, X and
Y follow Riesz distributions with the same scale parameter . In general, we
do not know whether a = should always belong to .
be a mapping defined through
Then, (U, V ) = (X, Y ). The inverse mapping 1 : D
(x, y) = 1(u, v) = (w(v)u, w(v)(e u)),
hence is a bijection. We are looking for the Jacobian of the map 1, that is, the
determinant of the linear map
dx/du dx/dv
dy/du dy/dv
J =
= Det(w(v)).
Det (w (v)) = (det v)dim / r .
Now, we can find the joint density of (U, V ). Since (X, Y ) have independent
components, we obtain
f(U,V )(u, v) = (det v)dim / r f X (w(v)u) fY (w(v)(e u))
E, C X , CY R and wmultiplicative functions
f(U,V )(u, v) = (det v)dim / r f X (w(v)u) fY (w(v)(e u))
= C1C2 (det v)dim / r e(w(v)u) f (w(v)(e u))
e , v I (w(v)u)I (w(v)(e u))
= C1C2 (det v)dim / r e(v) f (v)e , v I (v)
e(w(e)u) f (w(e)(e u))ID(u),
= fU (u) fV (v),
what completes the proof.
To prove the characterization of given measures, we need to show that the inverse
implication is also valid. The following theorem generalizes results obtained in
Bobecka and Wesoowski, Hassairi et al. and Koodziejek [2,11,13]. We consider
quotient U for any multiplication algorithm w satisfying conditions (i) and (ii) given
at the beginning of this section (note that multiplication algorithms w1 and w2 defined
in (10) and (11), respectively, satisfy both of these conditions). Respective densities
are then expressed in terms of wmultiplicative Cauchy functions.
Theorem 4.2 (The LukacsOlkinRubin theorem with densities on symmetric cones)
Let X and Y be independent rvs valued in irreducible symmetric cone with strictly
positive and continuous densities. Set V = X + Y and U = g (X + Y ) X for any
multiplication algorithm w = g1 satisfying conditions (i) and (ii). If U and V are
independent, then there exist E and wmultiplicative functions e, f such that
(26) holds.
In particular,
Proof We start from (27). Since (U, V ) is assumed to have independent components,
the following identity holds almost everywhere with respect to Lebesgue measure:
(det(x + y))dim / r f X (x) fY (y) = fU (g (x + y) x) fV (x + y),
where f X , fY , fU and fV denote densities of X , Y , U and V , respectively.
Since the respective densities are assumed to be continuous, the above equation
holds for every x, y . Taking logarithms of both sides of the above equation (it is
permitted since f X , fY > 0 on ), we get
a(x) + b(y) = c(x + y) + d (g (x + y) x) ,
a(x) = log f X (x),
b(x) = log fY (x),
c(x) = log fV (x) dimr
d(u) = log fU (u),
for x and u D.
The first part of the conclusion follows now directly from Theorem 3.6. Thus, there
exist constants E, Ci R, i {1, 2} and wlogarithmic functions e and f such
that
f X (x) = ea(x) = eC1 e(x)e , x ,
fY (x) = eb(x) = eC2 f (x)e , x ,
for any x .
Let us observe that if w(x) = w1(x) = P(x1/2), then for Theorem 3.1, there exist
constants i , i = 1, 2, such that e(x) = (det x)1 and f (x) = (det x)2 . Since f X
and fY are densities, it follows that a = , ki = pi (dim )/ r > 1 and
eCi = (det(a)) pi / ( pi ), i = 1, 2.
Analogously, if w(x) = w2(x) = tx, then Theorem 3.2 and Remark 3.3 imply that
there exist constants si = (si, j )rj=1, si, j > ( j 1)d/2, i = 1, 2, and a =
such that X Rs1,a i Y Rs2,a.
Remark 4.3 Fix k K and consider w(k)(x) = w(x)k. The w(k)multiplicative
function f satisfies equation
Substituting y k1y
f (x) f (w(e)ky) = f (w(x)ky).
f (x) f (w(e)y) = f (w(x)y),
that is w(k)multiplicative functions are the same as wmultiplicative functions. This
leads to the rather unsurprising observation that if we consider Theorem 4.2 with
w(x) = P(x1/2)k or w(x) = txk, regardless of k K , we will characterize the same
distributions as in points (1) and (2) of Theorem 4.2.
With Theorem 4.2, one can easily reprove original LukacsOlkinRubin theorem
(version of Olkin and Rubin [22] and Casalis and Letac [7]), when the distribution of
U is invariant under a group of automorphisms:
fU (u) = C e(w(e)u) f (e w(e)u).
The distribution of U is invariant under K , thus density fU is a K invariant function,
that is fU (u) = fU (ku) for any k K . Note that w(e) K , thus
e(u) f (e u) = e(ku) f (e ku), (k, u) K D.
We will show that both functions e and f are K invariant. Recall that e(x) e(w(e)y)
= e(w(x)y), therefore after taking y = e, we obtain e(x) = e(x)e(e) for any
> 0 and x . Inserting u = v into (30), we arrive at
Thus, e(v) f (e v) = e(kv) f (e kv) for any (0, 1], v D. Since f (e) = 1
and f is continuous on , by passing to the limit as 0, we get that e is K invariant
and so is f . By Theorem 3.4 and continuity of e and f , we get that there exist constants
1, 2 such that e(x) = (det x)1 and f (x) = (det x)2 ; hence, X and Y have Wishart
distributions.
Acknowledgments The author thanks J. Wesoowski for helpful comments and discussions. This research
was partially supported by NCN Grant No. 2012/05/B/ST1/00554.
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