#### The Matsumoto–Yor Property and Its Converse on Symmetric Cones

The Matsumoto-Yor Property and Its Converse on Symmetric Cones
Bartosz Kołodziejek 0
0 Faculty of Mathematics and Information Science, Warsaw University of Technology , Pl. Politechniki 1, 00-661 Warsaw , Poland
The Matsumoto-Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As was observed in Letac and Wesołowski (Ann Probab 28:1371-1383, 2000), the natural framework for the multivariate MY property is symmetric cones; however, they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we prove the converse to the symmetric cone-variate MY property, which extends some earlier results. The smoothness assumption for the densities of respective variables is reduced to continuity only. This enhancement was possible due to the new solution of a related functional equation for real functions defined on symmetric cones.
Matsumoto-Yor property; Generalized inverse Gaussian distributions; Wishart distributions; Symmetric cones; Hua's identity; Functional equation
1 Introduction
Matsumoto and Yor [15, 16] have shown that if X and Y are independent random
variables, Y is gamma distributed with the shape parameter p and the scale parameter
a and X has the generalized inverse Gaussian distribution (GIG) with parameters
(− p, a, b), then the random variables U = ( X + Y )−1 and V = X −1 − ( X + Y )−1
are independent with respective distributions GIG with parameters (− p, b, a) and
gamma with parameters p and b.
Matsumoto and Yor asked about the converse theorem based on the independence of
U and V . Assume that X and Y are non-degenerate nonnegative independent random
variables, such that U and V are independent. Does this imply that X and Y must
follow GIG and gamma distributions, respectively?
A positive answer to this question was given by Letac and Wesołowski [13], with
the use of Laplace transforms. In the same paper, both the Matsumoto–Yor property
and its converse (with additional smoothness assumptions) were generalized to the
cone + of symmetric positive definite (r, r ) real matrices in the following way. For
p > (r − 1)/2 and a, b ∈ +, consider two independent random variables X and Y
with following densities
μ− p,a,b(dx) = c1(det x)− p−(r+1)/2 exp −tr (a · x) − tr (b · x−1) I + (x)dx,
γ p,a(dy) = c2(det y) p−(r+1)/2 exp(−tr (a · y))I + (y)dy.
The distribution of X is the GIG with parameters (− p, a, b), and the distribution of
Y is the Wishart distribution with shape parameter p and scale parameter a. Letac
and Wesołowski have shown that if X and Y are as above, then (U, V ) has
distribution μ− p,b,a ⊗ γ p,b. As was observed by the authors, the natural framework for
Matsumoto–Yor property is symmetric cones. Statement of a symmetric cone version
of Matsumoto–Yor property is given in Sect. 3.
In this paper, we give a new proof of the converse result of the Matsumoto–Yor
property, when X and Y take values in any irreducible symmetric cone. The
smoothness assumption is reduced from C 2 densities in [13] and differentiability in [17] to
the continuity only. A new solution of a related functional equation on symmetric
cones (see Theorem 4.5) was found under the assumption of continuity of respective
functions with the use of the corresponding univariate result due to Wesołowski [18].
Similar reduction in regularity assumptions was recently performed in the density
version of Lukacs–Olkin–Rubin in [7].
It is worth mentioning that several related one-dimensional results [3,10] as well
as results for random matrices [9,12].
While solving the functional equation, we use Hua’s identity, which allows to write
the inverse of V = X −1 − (X + Y )−1 in a very convenient form:
V −1 = X + X · Y −1 · X.
Hua’s identity has already proved to be useful in some problems related to GIG and
Wishart distributions—see [1], where it was used to analyze some random continued
fractions on symmetric cones.
The paper is organized as follows. We start in the next section with some basic
definitions and theorems regarding analysis on symmetric cones. In Sect. 3, we define
the GIG and Wishart distributions and state the Matsumoto–Yor property on symmetric
cones. A core of the proof of the converse to the Matsumoto–Yor property is a solution
of some functional equation for real functions with arguments from the cone. Section 4
is devoted to analysis of this functional equation. The statement and the proof of the
2 Symmetric Cones
main result are given in Sect. 5. Finally, in Sect. 6, we give some remarks regarding
the MY property on matrices of different dimensions and related functional equation.
In this section, we give a short introduction to the theory of symmetric cones. For
further details, we refer to [4].
A Euclidean Jordan algebra is a Euclidean space E (endowed with the scalar
product denoted by 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:
L(x)y = xy,
P(x) = 2L2(x) − L x2 .
Let End(E) denote the space of endomorphisms of E. The map P : E → End(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 it is denoted by y = x−1. 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 x−1 .
A 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 denote either the
real numbers R, the complex ones C, the quaternions H or the octonions O. Let us
write Sr (K) for the space of r × r Hermitian matrices valued in K, endowed with the
Euclidean structure x, y = Trace (x · y¯) and with the Jordan product
xy = 21 (x · y + y · x),
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 if K = O, then
P(y)x = y · x · y.
(x0, x1, . . . , xn ) (y0, y1, . . . , yn ) =
i=0
To each Euclidean simple Jordan algebra, one can attach the set ¯ of Jordan squares
¯ =
x ∈ E : there exists y in E such that x = y2 .
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 symmetric cone of some Euclidean
simple Jordan algebra. Each simple Jordan algebra corresponds to a symmetric cone;
hence, there exists 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 (2) is called the Lorentz cone.
We will now introduce a very useful decomposition in E, called the spectral
decomposition. An element c ∈ E is said to be a primitive idempotent if cc = c = 0 and if c
is not a sum of two non-null idempotents. A complete system of primitive orthogonal
idempotents is a set (c1, . . . , cr ) such that
i=1
ci = e and ci c j = δi 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 λi ci for some complete
system of primitive orthogonal idempotents (c1, . . . , cr ). The real numbers λi , i =
1, . . . , r are the eigenvalues of x. One can then define the trace and the determinant
of x by, respectively, tr x = ri=1 λi and det x = ri=1 λi . An element x ∈ E belongs
to if and only if all its eigenvalues are strictly positive.
Note that up to a multiplicative constant, tr (xy) is the only scalar product on E
which makes self dual. Henceforth, we assume that is an irreducible cone and
that corresponding Jordan algebra E is equipped with canonical scalar product x, y =
tr (xy).
The rank r and dim of irreducible symmetric cone are connected through the
relation
= r +
where d is an integer called the Peirce constant.
The important property of the determinant is that
det (P(x)y) = (det x)2 det y, (x, y) ∈
It turns out that (3) characterizes determinant—see Lemma 4.2 below. Moreover (see
[4, Proposition II.4.2])
Det (P(x)) = (det x)2 dim /r ,
where Det denotes the determinant in the space of endomorphisms on .
In the proof of our main theorem, we will need the following identity (called Hua’s
identity—see [4, Exercise 5c, p.39])
a−1 − (a + b)−1 = (a + P(a)b−1)−1
when a ∈ , b ∈ E are such that b, a + b and a + P(a)b−1 are invertible. Note that if
a, b ∈ , then a−1 − (a + b)−1 ∈ . For the cone + of symmetric positive definite
real matrices, Hua’s identity takes the form given in (1).
3 Wishart and GIG Distributions
and any p in the set
= {0, d/2, d, . . . , d(r − 1)/2} ∪ (d(r − 1)/2, ∞)
by its Laplace transform
which holds for any σ +a ∈ . If p > dim /r −1, then γ p,a is absolutely continuous
with respect to the Lebesgue measure and has the density
(det x) p−dim /r e− a,x I (x) dx, x ∈
where is the gamma function of the symmetric cone (see [4, p.124]).
The absolutely continuous generalized inverse Gaussian distribution μ p,a,b on
is defined for a, b ∈ and p ∈ R by its density
(det x) p−dim /r e− a,x − b,x−1 I (x) dx, x ∈
where K p(a, b) is a normalizing constant.
In [13], Theorem 3.1 was proved in the special case of the cone of symmetric
positive definite real matrices +. As it was observed by the authors, symmetric cones
are the natural framework for considering the Matsumoto–Yor property. We state the
following theorem without a proof as it only mimics the argument for +. The original
proof relies on the properties of Bessel-like functions (K p(a, b)) introduced in [5],
which retain their usual properties in the symmetric cone setting.
Theorem 3.1 Let p ∈ and a and b in irreducible symmetric cone . Let X and
Y be independent random variables in and ¯ with respective distributions μ− p,a,b
and γ p,a. Then random variables U = (X + Y )−1 and V = X −1 − (X + Y )−1 are
independent with respective distributions μ− p,b,a and γ p,b.
4 Functional Equations
At the beginning of this section, we state three results that will be useful in the proof
of the main technical result—Theorem 4.5. The first one regards regular additive
functions (see [11]) on symmetric cone.
f (x) + f (y) = f (x + y), (x, y) ∈
Then there exists f ∈ E such that f (x) = f, x for any x ∈
An elementary proof of this theorem may be found in [6]. The following lemma
was recently proved in [8].
Lemma 4.2 (Logarithmic Pexider functional equation) Let f1, f2, f3 :
measurable functions such that
f1(x) + f2(y) = f3 P x1/2 y , (x, y) ∈
Then there exist a constant q ∈ R and constants γ1, γ2 ∈ R such that for all x ∈
The main technical result will rely on the following univariate result due to
Wesołowski [18].
Theorem 4.3 Let A, B, C and D be locally integrable real functions defined on
(0, ∞) such that
g(x (x + y)) − g(y(x + y)) = α(x ) − α(y), (x , y) ∈ (0, ∞)2.
Then there exist real numbers A, B, C and D such that for any x > 0,
g(x ) = Ax + B log x + C, α(x ) = Ax 2 + B log x + D.
The following result then follows from Theorem 4.3.
Theorem 4.4 Let A, B, C and D be locally integrable real functions defined on
(0, ∞) such that
A(x ) + B(y) = C (x + y)−1
+ D x −1 − (x + y)−1 , (x , y) ∈ (0, ∞)2.
A(x ) = − p log x + f x + gx −1 + C1,
B(x ) = p log x + f x + C2,
C (x ) = − p log x + gx + f x −1 + C3,
D(x ) = p log x + gx + C4,
and C1 + C2 = C3 + C4.
Proof Denote g1(x ) = A(x −1) − B(x −1) and α1(x ) = D(x 2). Interchange the roles
of x and y in (7) and subtract from the original equation. Then
g1 x −1
− g1 y−1
Inserting x = (u(u + v))−1 and y = (v(u + v))−1, we arrive at (6) with g and α
replaced, respectively, with g1 and α1.
Substituting x → (x + y)−1 and y → x −1 − (x + y)−1 in (7), we obtain
A (x + y)−1
+ B x −1 − (x + y)−1
= C (x ) + D(y), (x , y) ∈ (0, ∞)2.
As before, denoting g2(x ) = C (x −1) − D(x −1) and α2(x ) = B(x 2) and subtracting
the same equation with x and y interchanged, we see that (6) holds true for g2 and α2
also. Functions gi and αi , i = 1, 2, are locally integrable, because for g1 we have
| A(x −1) − B(x −1)| dx =
| A(y) − B(y)| dy
for all compact sets K ⊂ (0, ∞), where φ (K ) is the (compact) image of K under
φ (x ) = x −1. Since A and B were assumed to be locally integrable, we see that g1 is
locally integrable. Analogously, we proceed for g2, α1 and α2. Thus, by Theorem 4.3,
we obtain (we borrow this notation from Theorem 4.3):
B(x ) = α2(√x ) = A2x + B2/2 log x + D2,
D(x ) = α1(√x ) = A1x + B1/2 log x + D1,
A(x ) = A(x ) − B(x ) + B(x ) = g1(x −1) + α2(√x )
= A2x + A1x −1 − (B1 − B2/2) log x + C1 + D2,
C (x ) = C (x ) − D(x ) + D(x ) = g2(x −1) + α1(√x )
= A1x + A2x −1 − (B2 − B1/2) log x + C2 + D1,
Inserting it back into (7), it can be quickly verified that B1 = B2 = B.
We are now ready to state and solve the functional equation related to the Matsumoto–
Yor property on symmetric cones.
Theorem 4.5 Let a, b, c and d be continuous real functions defined on
a(x) + b(y) = c (x + y)−1
+ d x−1 − (x + y)−1 , (x, y) ∈
a(x) = q log det x + f, x + g, x−1 + γ1 + γ3,
c(x) = q log det x + g, x + f, x−1 + γ3,
Proof By inserting (x, y) = (αz, βz) for α, β > 0 and z ∈ into (8), we arrive at the
equation (7) with A(α) := a(αz), B(α) := b(αz), C (α) := c(αz−1) and D(α) :=
d(αz−1). Functions A, B, C and D are continuous, so they are locally integrable.
Therefore, by Theorem 4.4, for any z ∈ , there exist constants p(z), f (z), g(z) and
Ci (z), i = 1, . . . , 4, such that
a(αz) = − p(z) log α + f (z)α + g(z)α−1 + C1(z),
c(αz−1) = − p(z) log α + g(z)α + f (z)α−1 + C3(z),
d(αz−1) = p(z) log α + g(z)α + C4(z),
C1(z) + C2(z) = C3(z) + C4(z),
for any α > 0 and z ∈ . Functions z → p(z), z → f (z), z → g(z) and z →
Ci (z), i = 1, . . . , 4, are continuous, because a, b, c and d are continuous. Let β > 0.
By the equality a(α(βz)) = a((αβ)z), we obtain that for any α > 0,
a(αβz) = − p(z) log αβ + f (z)αβ + g(z)α−1β−1 + C1(z)
= − p(βz) log α + f (βz)α + g(βz)α−1 + C1(βz),
Following the same procedure for functions b, c and d, we have
+ g (x−1 − (x + y)−1)−1
+ C4 (x−1 − (x + y)−1)−1 .
Consider the above equation for (α−1x, α−1y) ∈
α−1 f (x) + αg(x) + C1(α−1x) + α−1 f (y) + C2(α−1y)
= αg(x + y) + α−1 f (x + y) + C3(α−1(x + y))
+ C4 α−1(x−1 − (x + y)−1)−1 .
Multiplying both sides of the above equation by α and passing to the limit as α → 0,
by (11), we obtain
f (x) + f (y) − f (x + y)
= αli→m0 α C3(α−1(x + y)) + C4 α−1(x−1 − (x + y)−1)−1
By (10) and (11), the limit on the right-hand side of the above equation equals 0. Thus,
by Lemma 4.1, there exists f ∈ E such that f (x) = f, x . Analogously, consider (12)
for (αx, αy) ∈ 2, α > 0, multiply its both sides by α and pass to the limit as α → 0.
Then
Define g¯(x) = g(x−1). Then,
g(x) − g(x + y) − g (x−1 − (x + y)−1)−1
= αli→m0 α C3(α(x + y)) + C4 α(x−1 − (x + y)−1)−1
g¯(x−1) = g¯((x + y)−1) + g¯(x−1 − (x + y)−1).
Thus, g¯ is additive, i.e., there exists g ∈ E such that g(x) = g, x−1 .
By the use of above results for f and g, (12) simplifies to
C1(x) + C2(y) = C3(x + y) + C4 (x−1 − (x + y)−1)−1 .
Recall that by Hua’s identity (5), the argument of C4 above may be written as
(x−1 − (x + y)−1)−1 = x + P(x)y−1.
= C1(x) + C2(αz) = C3(x + αz) + C4 α−1(αx + P(x)z−1)
= C3(x + αz) + C4(αx + P(x)z−1) + p(αx + P(x)z−1) log α.
C1(x) + C2(z) − C3(x) − C4(P(x)z−1) = αli→m0 log α p αx + P(x)z−1
− p(z)
2. A necessary condition for the limit on the right-hand side to exist
= 0.
But p is continuous and limα→0 p(αx + P(x)z−1) = p(P(x)z−1), hence p(z) =
p(P(x)z−1). Thus, function p is constant and the right-hand side of (14) is equal to
0. Hence, substituting z = y−1 and x → x1/2 in (14), we get
C1(x1/2) − C3(x1/2) + C2(y−1) = C4(P(x1/2)y).
Define f1(x) := C1(x1/2) − C3(x1/2), f2(x) := C2(x−1) and f3(x) := C4(x) for
x ∈ . Then
f1(x) + f2(y) = f3(P(x1/2)y), (x, y) ∈
Let us go back to (13) and use the above result. Then
C3(x) + 2q log det x − q log det y
= C3(x + y) + q log det(x + P(x)y−1), (x, y) ∈
Since det(x + P(x)y−1) = det(x2) det(x−1 + y−1), we obtain
C3(x) − q log det y = C3(x + y) + q log det x−1 + y−1 .
One can interchange x and y on the right-hand side to obtain
C3(x) + q log det x = C3(y) + q log det y = const := γ3,
5 Main Result
In the following section, we prove our main result, which is a converse to the
Matsumoto–Yor property in the symmetric cone-variate case. We reduce the
smoothness conditions for densities from C 2 densities in [13] and differentiability in [17] to
the continuity only.
Theorem 5.1 Let X and Y be independent random variables in with continuous
and strictly positive densities. If the random variables U = (X + Y )−1 and V =
X −1 − (X + Y )−1 are independent, then there exists p > dim /r − 1, a and b in
such that X and Y follow respective distributions μ− p,a,b and γ p,a.
Proof Define the map : 2 → 2 by (x, y) = (x + y)−1, x−1 − (x + y)−1 =
(u, v). Obviously, (U, V ) = (X, Y ). Function is a bijection. In order to find the
joint density of (U, V ), the essential computation is the one involved with finding the
Jacobian J of the map ψ −1, that is, the determinant of the linear map
dx/du dx/dv
dy/du dy/dv
It is easy to see that = −1, that is (x, y) = (u + v)−1, u−1 − (u + v)−1 . Note
that the derivative of the map x → x−1 is −P(x)−1. Thus
J =
−P(u + v)−1 −P(u + v)−1
−P(u)−1 + P(u + v)−1 P(u + v)−1
−P(u)−1 0
= −P(u)−1 + P(u + v)−1 P(u + v)−1
= Det P(u + v)−1P(u)−1 .
By (4), we get
J = (det u det(u + v))−2 dim /r .
Since (X, Y ) and (U, V ) have independent components, the following identity holds
almost everywhere with respect to the Lebesgue measure:
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
(u, v) ∈ 2. Taking the logarithms of both sides of the above equation (it is permitted
since f X , fY > 0 on ), we get
a(u) + b(v) = c (u + v)−1
+ d u−1 − (u + v)−1 ,
By Theorem 4.5, there exist constants q ∈ R, f, g ∈ E and γi ∈ R, i = 1, 2, 3, such
that for any x ∈ ,
a(x) = log fU (x) + 2 dirm log det x,
c(x) = log f X (x) + 2 dirm log det x,
b = log fV , d = log fY .
c(x) = −q log det x + g, x + f, x−1 + γ3,
f X (x) = eγ3 (det x)−q−2 dim /r e g,x + f,x−1 ,
6 Comments
Since f X and fY are some densities, we have a = −g ∈
q = p − dim /r > −1. Thus, X ∼ μ− p,a,b and Y ∼ γ p,a.
, b = −f ∈
Recall that Sr (K) denotes the space of r × r Hermitian matrices valued in K. Let
r (K) be the symmetric cone of Jordan algebra E = Sr (K), where K denotes either
the real numbers R, the complex ones C or the quaternions H. We exclude here the
non-associative case K = O.
Let z be a fixed s × r matrix of full rank valued in K and define the linear mapping
Psr : Sr (K) → Ss (K) by
Psr (z)x = z · x · z∗.
If r = s, then Psr is the ordinary quadratic representation of s . In the rest of the paper,
we will drop the subscript and simply write P (abusing the notation from previous
sections).
Now, consider the following transformation ψz : r (K) × s (K) → s (K) ×
r (K), where
ψz (x, y) = (P(z)x + y)−1, x−1 − P(z∗)(P(z)x + y)−1 .
It is natural to ask whether an analogue of Theorem 5.1 holds if we consider
independent random variables X and Y valued in r (K) and s (K) and define
(U, V ) = ψz (X, Y ). The answer is affirmative, and it was given in [14, Theorem 4.1].
Following the same steps as in the proof of Theorem 5.1, the problem of
characterization of probability measures is reduced to the problem of solving following functional
equation
a(u) + b(v) = c (P(z∗)u + v)−1
+ d u−1 − P(z)(P(z∗)u + v)−1 ,
(u, v) ∈
where a, d : s (K) → R and b, c : r (K) → R are some unknown functions. This
functional equation was solved by Massam and Wesołowski [14] for K = R under
the assumption that the unknown functions are differentiable. It can be shown that
through Theorem 4.5, this assumption may be weakened to continuity. Therefore, we
obtain the following refinement of [14, Theorem 4.1]:
Theorem 6.1 Let X and Y be independent random variables with values in r (K)
and s (K), respectively. Assume that X and Y have continuous densities, which are
strictly positive. Define (U, V ) = ψz (X, Y ).
If U and V are independent, then there exist matrices (a, b) ∈ s (K) × r (K)
and a constant p > dim r (K)/r − 1 such that
where q = p + (dim
s (K)/s − dim
The superscripts (s) and (r ) are used to emphasize the ranks of the cones on which the
distributions are considered.
The solution to (16) was also used in the proof of the characterization of Wishart
distribution through its block conditional independence structure (see [14, Theorem 5.1].
One of the technical assumptions was that the respective random matrix has a
differentiable density. This was assumed only in order to solve a functional equation, whose
solution was not known under weaker assumptions. Therefore, this assumption may
be reduced to the existence of continuous densities.
An analogous assumption was imposed on the densities in the recent paper of
Bobecka [2], where the multivariate MY property on trees is considered—see [2,
Theorem 4.3]. Thanks to the solution of (16) under weaker assumptions, this theorem
holds true if we assume continuity of densities only.
Acknowledgments This research was partially supported by NCN Grant No. 2012/05/B/ST1/00554.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
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