ON a-SYMMETRIC MULTIVARIATE DISTRIBUTIONS*
by
Stamatis Cambanis, Robert Keener t and Gordon Simons
University of North Carolina
ABSTRACT
A random vector is said to have a I-symmetric distribution if its
characteristic function is of the form ¢(Itll
+ ••• +
Itnl).
I-symmetric
distributions are characterized through representations of the admissible
functions ¢ and through stochastic representations of the random vectors,
and some of their properties are studied.
AMS 1970 Subject Classification:
Key Words and Phrases:
Primary 60EOS.
a- symmetric, multivariate, characteristic
function, Laplace transform, conditional
distribution, n-dimensional version, stable.
*This research was supported by the Air Force Office of Scientific
Research Grant AFOSR-80-0080 and by the National Science Foundation
Grants MCS78-0l240 and MCS8l-00748.
t Now at the University of Michigan.
FOOTNOTES
1 This proof is due to Hari Mukerj ee.
We are aware of several other
proofs, including one of our own and two distinct geometric proofs
produced by Norman L. Johnson and (again) by Mukerjee.
2 For the remainder of this section, Xl and X will denote vector's
2
rather than single components of X, and in Theorems 4.1 and 4.2 below,
they will denote subvectors of X.
.e
1
1.
INTRODUCTION
We shall say that a random vector X = (Xl' .•. ' Xn ) has an
a-symmetpic distPibution (a > 0) if its characteristic function is of the
form
(1.1)
for all t = (tl, •.. ,tn )
E
n
lR, and we shall write X....., S (a,¢).
n
also denote by cI>n (a) the class of all functions ¢: [0,00)
such that <pc!tl,a
+ ... +
-+
We shall
lR which are
n
Itn,a) is a characteristic function on lR • We
are interested here in studying a-symmetric distributions through
representations of the classes cI>n (a), through stochastic representations
of the random vectors X, and, where feasible, through representations of
their probability density functions.
It is clear that a-symmetric distributions in one dimension are
simply symmetric distributions, and that the marginals of a-symmetric
distributions are a-symmetric.
cI>n (a) are nonincreasing in n.
It follows, for given a, that the classes
We shall let cI>00 (a) = nnn
cI> (a) so that
cI>n (a) {- cI>00 (a) as n -+ 00 •
2-symmetric distributions are naturally called spherically symmetric,
and have been studied extensively.
¢
E
cI>n (2) if and only if
¢(u)
.e
Schoenberg (1938) showed that
u ;::: 0 ,
for some distribution function F on [0,00), where
(1.2)
Q
n
(t 12 + ••• + t n2) is the
characteristic function of a random vector U = (Up •.• , Un) which is
2
unifonnly distributed on the surface of the unit sphere in lRn .
It
follows immediately that X has a spherically symmetric distribution,
i.e., X
~
Sn(2,¢) if and only if it has the stochastic representation
X ~ RU ,
where R
~
(1.3)
° is independent of U and has distribution function F.
This
makes precise the intuitively clear property that spherically symmetric
distributions are scale mixtures of unifonn distributions.
In this paper, we establish integral representations of ¢
analogous to (1.2), and stochastic representations of
I-symmetric distributions analogous to (1.3).
XIS
E
~n(l)
with
Again, I-symmetric
distributions are scale mixtures of a specified "primitive distribution".
We expect that analogous representations hold for other values of a
besides 1 and 2.
We have some evidence for this in the case a
= ~
(n = 2).
The classes
00 (a) have been studied by Bretagnolle, Dacunha Castelle
~
and Krivine (1966), and
¢
¢
00 (2) by Schoenberg (1938):
For a
E
(0,2],
00 (a) if and only if
E ~
~() =
't' u
J[0,00)
a
e- ur dF(r) ,
for some distribution function F on [0,00).
u ~
°,
Equivalently, X is a finite
segment of (or) an infinite dimensional vector whose finite dimensional
distributions are a-symmetric if and only if
.e
X
~
RY
,
3
e
where R ~
a
is independent of Y which has independent and identically
distributed symmetric stable components of index a and characteristic
functions E exp(itYk) = exp(-Itl a ). I.e., a-symmetric infinite
sequences of random variables are scale mixtures of independent and
identically distributed stable variables of index a (a consequence of
deFinetti's theorem).
When X '" Sn (a,<p), its component random variables X are identically
k
distributed with X '" Sl(a,<p). It follows from (1.1) that
k
(1.4)
Recently, Eaton (1981) introduced the interesting notion of n-dimensional
versions of one-dimensional symmetric distributions:
The distribution of
X is an n-dimensional version of the distribution of a symmetric random
variable Z if t 1X1 + ••• + tnXn ~ c(t) Z for t = (t l , ... , t n ) E :IRn with
c(t) ~ a for all t. Relationship (1.4) shows that the distribution of X
is an n-dimensional version of the distribution of Xl with
c(t) = (Itlla
+ ••• +
Itnla)l/a.
Thus the results of this paper provide
specific constructions of some n-dimensional versions of distributions
with characteristic functions of the form <PC It \) (t
(1).
n
However, these n-dimensional versions may constitute a small subclass of
all n-dimensional versions.
<p(u)
= exp(-u)
(<p
E
<poo(a),
E
R), <p
E
<P
For instance, in the special case
a<a
~
2), the n-dimensional versions of the
symmetric stable distribution with index a defined through (1.1) have
independent and identically distributed stable components of index a,
.e
while, as shown in Theorem 6.1, the class of all n-dimensional versions
consists of all symmetric n-dimensional stable distributions of index a.
(Half of this theorem is due to Eaton (1981).)
4
We do not know whether the classes <P (a), 2 ::; n < 00, contain
n
nondegenerate members (<P F 1) when a > 2. They clearly do when n = 1,
and do not when n = 00
(See Bretagnolle, Dactmha Castelle and Krivine
(1966).)
The paper is organized as follows:
For a
= 1,
we consider in
Section Z the bivariate case; in Section 3 the general multivariate case
n
~
2; and in Section 4 certain properties of I-symmetric distributions,
including a description of their conditional distributions, which are
also I-symmetric.
cases n
We believe that clarity is enhanced by treating the
2 and n > 2 separately; there is virtually no overlap between
=
The case "a = !:2 and n = Z" is discussed (but not
Sections 2 and 3.
fully resolved) in Section 5.
Finally, n-dimensional versions and a
possible generalization of a-symmetry are discussed in Section 6.
It should be evident that much work remains to be done on
a-symmetric distributions when a
2.
~
1, Z.
BIVARIATE l-SYM-ffiTRIC DISTRIBUTIONS
In this section, we derive integral representations of functions in
<PZ(l), and stochastic representations of bivariate random vectors with
I-symmetric distributions.
Such distributions are absolutely continuous,
except possibly at zero; we derive, as well, representations of their
densities.
TI-IEOREM 2.1.
.e
(a)
<P
€
<P Z(1) if and only if
<P(u) = J[O,oo) <PO(ru)dF(r) ,
u
~
0
,
where F is a distribution function on [0,00) and <PO
(Z.l)
€
<PZ(l) assumes the
5
e
form
sin v
dv ,
v
(2.2)
u ;::: 0 .
Expressed more simply,
¢(u) = ~
7f
(b)
fX' sin uv F(v)dv,
v
JO
U
(2.3)
> 0 •
Equivalently, the characteristic function of (X,Y) has the
foY'l'Tl
Eei(sX+tY)
= ¢(Isl +Itl)
,
(2.4)
S,tElR,
if and only if
(2.5)
where R is a nonnegative random variable with distribution function F,
and (X ' yO) is independent of R and has the characteristic function
O
I I I I) .
¢O ( s + t
A stochastic description of (X O' Y0)
( X Y)
0' 0
k
is given by
(U
V 1 )
B~ '(I-B)~ ,
(2.6)
2
where (U , V) is unifOY'l'Tlly distributed on the unit circle of lR , and B is
distributed
(c)
Beta(~,~)
independently of (U,V).
Equivalently, a bivariate distribution is I-symmetric if and
2
only if it is absolutely continuous on lR - {(O,O)} with a density of
.e
the foY'l'Tl
6
g(x,y) --
f (0,00) r -z
x y
go (y'
y)dF(r)
(Z.7)
(and has an atom of size F(O) at (0,0)), where F is a distribution
function on [0,(0) and go is the bivariate density of (XO,Y ) given by
O
go(x,y)
=-[l,oo)(lxl) - 1[1,(0) (!yl)
=
222
7f (x - Y )
,
x Z ;f Y2
(Z.8)
Expressed more simply,
g(x,y)
=~ ,
x Z ;f YZ
(x - Y )
7f
We shall refer to
.
(Z.9)
(and occasionally to (XO,YO) and gO) as a
Part (a) of the theorem characterizes members of <P Z(1) as
primitive.
<PO
scale mixtures of a primitive.
The work of Schoenberg (1938) and of
Bretagnolle, Dacunha Castelle and Krivine (1966) referred to can be
viewed as providing the same kind of characterization of members of
<P
n
(a) for a
= Z, 1 ~
n
~
and for n
00,
= 00, 0 <
a < 2, respectively.
Theorem 3.1 and Section 5 below provide additional evidence that the
members of every class
primitive
<PO
E
<P
n
(a) are expressible as scale mixtures of a
<pn(a) (0 < a
~
2, 1 ~ n ~ (0).
In the course of the proof of Theorem 2.1, we will use the following
facts of independent interest contained in Propositions Z.l and Z.Z.
PROPOSITION Z.1.
aU s,t
.e
E
If the random variable B is
Beta(~ ,~),
then for
JR.,
s
Z
B
(Z.lO)
7
1 It is sufficient to consider the case s, t ~ 0 with
t > 0, and to let B = sin Z 8, where 8 is unifonnly distributed on the
Proof.
s +
interval (0, TI/Z). Then the left-hand side of (Z.lO) becomes
(sZ/sin Z 8) + (t Z/cos Z 8), which is expressible as (s +t) Z (1 + [T (8)] Z),
x
where x = (s -t)/(s+t) and Tx (8) = (x + cos Z8)/sin Z8. Thus it
suffices to verify that the distribution of ITx (8) I is independent of x
for -1 ~ x ~ 1, or, more specifically, that for all such x,
Pre ITx (8) I
~
z) = -TIZ tan -1 z,
z >
a.
(Z .11)
This is easily shown for x = ±l. When -1 < x < 1, Tx (8) is a strictly
decreasing function of 8 on (0, TI/Z) , and has range lR. Fix z, and let
80 and 81 > 80 be the two roots of ITx (8) I = z. Then
x + cos Z8 = z sin Z8 and x + cos Z8 = -z sin Z8 , which yield
0
0
l
l
cos Z8 - cos Z8 = z(sin Z8 + sin Z8 ) and, in turn, 8 - 8 = tan- l z,
0
l
0
l
1
0
from which (Z.ll) follows.
0
Proposition Z.1 is equivalent to a curious definite integral
fonnula:
which is valid for all s
integrals make sense.
~
0, t
~
0, and all functions f for which the
This surprisingly general identity appears to be
new.
.e
PROPOSITION Z. Z.
circle in R
Z
Let (D, V) be unifo'PlTlly distributed on the unit
and B be distributed
Beta(~,~)
independently of (D, V).
8
Then
=
(+,
B~
v
(I-B)~
)
(2.12)
has characteristic function
sin v
dv "
v
(2.13)
S,tElR"
joint density
1
for
Ix I <
1 :s; Iy I
or Iy I
< 1 :s;
IxI "
(2.14)
o
•
otheruJise "
and common marginal densities
u
~
0 .
(2.15)
Moreover"
s,t
Proof.
E
lR •
(2.16)
(2.14) follows from (2.12) by a straightforward computation.
Likewise, (2.15) follows from (2.14) by a straightforward, albeit messy,
computation.
shown.
.e
(2.16) follows easily from (2.13), which remains to be
The characteristic function of (U,V) is well-known (see, for
instance, Schoenberg (1938)) and is given by E exp[i(aU+bV)] =
2 2 k
J ( [a + b ] 2), where J
O is a Bessel function of the first kind.
O
obtain by (2.12) and Proposition 2.1,
Thus we
9
Ee
i(sXO+tY O)
2
t2
1
1 ... 1...
I+t I
= EJO([sB + -]~) = EJO(~)
B~
1-B
•
Putting Is I + It I = a and using the identity JO(ax) = 1 - x foa J 1 (vx)dv
(cf., Gradshteyn and Ryzhik (1980), 6.511 (7)), we have (with the help
•
of Fubini's theorem):
=
1:. ~1
7T
2
= TI
0
roo
11
J ( ~)
dx
= ~
0 IX /x(l-x)
7T
dx
r-z-
x/x"-l
=1 _
2 ra
r,1 J 0(ax) rzdx
x/x~-l
r,
dx
TI )0 dv 1 J 1 (vx) -2-
IT;
ra sin v d _ 2 roo sin v d
7T)0
v
v - TI la
v
v
~
•
4It
(cf., Gradshteyn and Ryzhik (1980), 6.552 (6), 8.464 (1,2), 8.465 (1),
0
3.721 (1)).
Proof of Theorem 2.1.
the form (2.4).
Let the characteristic function of (X, Y) have
We will show (2.1), (2.5) and (2.7).
2
Assume first that the characteristic function is IR -integrab1e:
(2.17)
Then (X,Y) has a continuous density f(x,y) given by
.4It
10
g(x,y)
1
=
4'IT
2
1
f;
= 2
•
roo roo e-i(sx+ty)
<Hlsi +It\)dsdt
~ cos(sx)cos(ty)¢(s+t)dsdt
'IT
fa du¢(u)
1
= -2
'IT
f~ dt cos([u - t]x)cos(ty)
¢eu) x sin(ux) - y sin(uy) du for x 2
= 12 C
o
2
2
'IT
X
~
2
Y
- Y
which is just the right side of (2.9) with the function F defined by
F(x)
•
Since g
2:
=
x ~ sin(ux)¢(u)du ,
X 2:
0 .
0, (2.9) implies that F is nondecreasing.
(2.18)
Thus the right side
of (2.7) (with this F and go defined in (2.8)) is well-defined, and
•
equality (2.7) can be viewed as an alternative way of writing (2.9).
Since g and go are p. d. f. 's, we have
1 =
=
roo roo g(x,y)dxdy
f (0,00)
dF(r)
roo roo r- 2 go(~ ) ~)dxdy = f (0,00)
and consequently F is a d.f. on [0,00).
dF(r) ,
This shows (2.7); (2.1) and
(2.5) follow from Proposition 2.2.
Now let ¢ in (2.4) be any member of <1>2(1) and let
¢ (u)
Then ¢ satisfies (2.17), and
n
-1
-1
the conclusions of the theorem hold for (X + n W, Y+ n Z), where W and
n
=
¢(u)exp( -Iul/n), u
2:
0, n
2:
1.
Z are independent Cauchy variables which are jointly independent of
(X,Y).
Thus
11
for some Rn ~ 0 which is independent of (Xo' YO), n ~ 1. From this,
(2.5) is easily established, and then (2.1) and (2.7) follow from
Proposition 2.2.
Conversely, if any of the equivalent relations (2.1), (2.5) or (2.7)
holds, then (2.4) follows from Proposition 2.1 in the same way that
o
(2.13) follows from (2.12).
The correspondence between ¢
2(1) and the distribution function
Indeed, F determines ¢ uniquely by (2.1).
F in (2.1) is one-to-one.
<P
E
Conversely, if ¢ satisfies (2.17), then F is determined by (2.18), and,
•
•
in general, F is the weak limit of {Fn}, where
Fn(X) = x ~ sin(ux)¢(u)exp(-u/n)du, x ~ 0, n ~ 1.
Alternatively, one
can derive (using (2.18) and a simple weak convergence argument) the
A
following general form of the Lap1ace-Stieltjes transform F of F (and
thereby establish the uniqueness of F):
roo
A
pes) = 10 (1 +v)
-2
<p (slV )dv
,
s
~
0 .
(2.19)
The following theorem based upon (2.19) is analogous to Theorem 2 of
Cambanis, Huang and Simons (1981), and its proof is given in Section 3
in a more general setting.
TIIEOREM 2.2.
A
bounded continuous function ¢ de fined on [0,00) is a
member of <P (1) if and only if the right-hand side of (2.19) is the
2
Laplace
transfo~
of a nonnegative random variable.
12
3.
MULTIVARIATE I-SYMMETRIC DISTRIBlITIONS
In this section, we continue the study of I-symmetric distributions
by deriving the analogue of Theorem 2.1 for n > 2.
Below, U = (U , ... , Un) denotes an n-dimensional random vector which
l
n
is uniformly distributed on the unit sphere in R , D = (D ,···, D )
I
n
denotes an n-dimensional random vector with Dirichlet distribution and
parameters
(~ , ... , ~), and SGn (ti
+... +
t~) is the characteristic
function of U, which is given by
~-l
= r(n/2)(Z/r)Z
I
n
--1
r > 0 .
(r)
Z
THEOREM 3.1.
¢
€
¢n(l) (n
~
2) if and only if
¢(u) = f[o,oo) ¢O(ru)dF(r) ,
(3.1)
whepe F is a distnbution function on [0,00) and ¢O
r
2
n
--
¢Z(l) is given by
€
n-3
-
SGn(XU)x Z (x-I) Z
dx
Equivalently, the joint chapactenstic function of X
(3.2)
=
(Xl"'" X ) has
n
the form
(3.3)
if and only if
13
(3.4)
•
where R is a nonnegative random variable with distribution function F,
and R, U = (U , ••• , Un)' and D = (D , ••• , D ) are independent.
1
1
n
In this
case, X (has an atom of size F(O) at 0 and) is absoluteZy continuous on
n
lR - 0 with density g given by
g(X) = f [0,(0) r -n goer-1 x)dF(r) ,
(3.5)
2
n
=
(n-2)!
'IT
2 n-2
f (O,lxkj)(xk-r)
L
n k=1
-n+2
r
dF(r)
n
2
2
II (x - xJo)
j=1 k
j;tk
k ;t j ,
where
(n-2)!
n
'IT
n {2
n 2 2
L (xk - 1) n-2
+ / IT (xk - x
k=l
j=l
J
0
) }
(3.6)
,
j;tk
k ;t j .
In proving Theorem 3.1, we will use the following higher dimensional
versions of Propositions 2.1 and 2.2.
PROPOSITION 3.1.
t~
Dn
L
For aU reaZ t l ,···, t n ,
(I tIl
+... + It n j) 2
D
l
(n ~ 2) •
(3.7)
14
PROPOSITION 3. 2.
The random vector
(3.8)
has characteristic function <PO ( ItIl + ••• +
It n I)"
where <PO is defined by
go described in
(3. 2)" and probabi li ty density function
(3.6) (n
~
2).
By Propositions 2.1 and 2.2, these claims are true when n = 2.
Their proofs for general n > 2 require induction, and are based upon the
following (easily justified) stochastic representations of D and U:
D
L
=
((l-D )D ' , ... ,(l-D )D' l'
n l
n n-
U
k
D)
n
= ((l-Dn )D',
D)
n
(3.9)
and
2
2
(/1_Un U' , ... , /1_Un U'n- l' Un )
l
=
2
(/1-U U' U)
n
'n '
(3.10)
where D' = CD1', ... , D'n- 1) is Dirichletn- 1C~"",~) and independent of
D, and U'
•
ln
n-1
m
~
=
CUi,,,,,
U~_l)
and'lndependent
is uniformly distributed on the unit sphere
0f
U
H
D D' , U and U' are
.
ereafter we assume,
jointly independent.
Proof of Proposition 3.1 for n > 2.
that (3.7) holds for n - 1
~
1.
Using this twice and using the symmetry
of the distribution of D, we obtain
.e
Make the induction hypothesis
15
+
k
_1_ •
o
l-D
n
Proof of Proposition 3.2 for n > 2.
Beta(~, (n-l)/2).
Putting
T
We need the fact that D is
n
= Itll + ... + It '
n
and Xo = (Xl'·'" Xn ), we
obtain from Proposition 3.1,
Ee
i (tlXl +... + t X )
i (tlXl + ... + t X) (n)
nn =EE{e
nn\D }
=
Turning our attention now to go' we asstnne that (3.6), with n
.e
replaced by n -1, gives the p.d. f. of X6 = (Xi,"" X~_l) =
(UVIDi , ... , U~_//D~_l)
for some n - 1 ~ 1.
To complete the
16
induction step, we must show that (3.6) is the p.d.f. of
or
Making the transformation
where
k=I, ... ,n-l,
x
n
=
whose Jacobian is d-~ [(1- u 2) / (1- d )] (n-l) /2, and using the facts that
n
n
n
D is Beta(~, (n-l)/2) and U has p.d.f.
n
n
-1 < u
n
< 1 ,
we find that the ((n+l)-dimensional) random vector (XO,D ) has the
n
p.d.f.
n-l
(n-3)!
.e
n
Tf
l
k=1
17
Integrating out the variable d , we obtain the p.d.f. of X :
n
O
n-1
(n-3)!
1
L
n
(3.11)
2
k=l n-1 2
IT (x - x.)
j=l k
J
7T
j~k
r2(~)
n-1 (x 2 _ 1)n-2
+
k
2
=
L n 2 2
n
(n-2)! 7T
k=l
IT (xk-x.)
j=l
J
(x 2 _ 1)n-2 n-1
n
+
L
x2 (n-2)
k=l
n
x
2(n-2)
k
n
2
2
IT (x - x.)
k
j=l
J
j~k
j~k
where
-2
[O,x ]
n
2
2
2
-2
[(1 - x ) / (x - x ), x ] for x 2 ~ 1 < x 2 '
k
n
n
k
n
k
I
kn
=
222
[0, (x - l)/(x - x )]
k
k
n
2
for x
n
o (the
2 2
for xk,x
n
empty set)
~
2
1 < xk '
~
1 .
The right side of (3.11) reduces to go' as described in (3.6),
since
2(n-2)
k
L
2
k=O nIT (x 2 -x.)
j=l k
J
n
x
=0
,
k
~
j .
j~k
The latter is seen as follows.
.e
2
n-2
With Yk = xk and hey) = y
, the sum
becomes the (n -1) - st divided difference of h at the points yl' •.. , yn'
which is zero, since the (n-1)-st derivative of h vanishes.
0
18
In tenus of a Bessel f1ID.ction, the primitive <PO
•
n-3
(r 2 _ 1) 2
]n_2(ur)
3n-4
dr,
~
For n = 3, ]1 (z) = (2/nz)
-
~
~
€
2
2
r
n
(l)
is given by
u > 0 (n
~
2) .
-2
sin z, and thus
1 sin u + ~ cos u - ~2 roo sin x dx,
2u
Ju
x
=
<P
Similar expressions for <PO can be obtained for n
corresponding expressions for
]k+~
=
u > 0 .
2k + 3 using the
(see Gradshteyn and Ryzhik (1980),
8.463).
Proof of Theorem 3.1.
Let the characteristic f1ID.ction of X have the
fonu (3.2), and assume first that it is IRn -integrab1e:
Then X has a continuous p.d.f. given by
g(x)
1
=
(2n)n
=
~ J
n
J n
m
e
-i(t 1x 1+···+ t x )
n n <p(lt1 ' +... + !tnl)dt1 ... dtn (3.13)
n cos (t 1x 1 ) ... cos(tnxn ) <P(t 1 +... + t n )dt 1 ··· dt
n
lR+
19
e
n
where Even h(xl'''.' xn ) = 2- I h(± Xl'.'" ± xn ), the sum being taken
over all 2n possible choices of the signs. If we denote by
[Yl' ... ,Y ; B(o)] the (n-l)-st divided difference of B(o) at the
n
(distinct) points Yl'··.' Yn ' we have
n
= I
k=l
(3.14)
n
n (Yk- Y.)
j=l
j;tk
J
and it easily can be shown by induction (under modest assumptions) that
(3.15)
(It is sufficient that a and B be measurable functions on lR+ and lR
respectively, that B have n - 2 absolutely continuous derivatives, and
that the left-hand integrand in (3.15) be Lebesgue integrable on lR~.)
It follows from (3.13) and (3.15) that
7T
n
_
roo
.-(n-l)
.
g(x) - Even JO <Huh
[Xl' ... ' xn ' exp(iu o ) ] du ,
and since
n-l
.[ 2
2
sin(ur-)],
1 Xl'···' xn ; (0) 2
Even[xl'''.'xn ; exp(iu o ) ] =
.e
we obtain
[xi,... ,x~;
n even ,
n-l
(0)-2- cos(ur-)] ,
n odd ,
20
e
IT
n
Z
Z
g(X) = [xl'· .. 'x ; Bn(o)]
n
(Ixjl ~ Ixkl, j
~
k)
,
(3.16)
where
n-Z
n-l
(o)-Z(-1) Z
-
fooo
sin(ur-)<p(u)du ,
n even ,
fa
cos(ur-)</>(u)du ,
n odd.
B (0) =
n
n-l
n-l
(0)-2(-1) 2
It follows from (3.lZ) that B
n
is (n -1) -times continuously differentiable
on (0,00) and, moreover, B(k) (0+), k = 1, ... , n-Z, exist and are finite.
In fact, from the nonnegativity of (3.16) (g is a p.d.f.) follow nearly
»
~
as strong assertions concerning B
n
and the nonnegativity of B(n -1) .
(See Roberts and Varberg (1973), page Z38.)
B(y)=
1
n
(n-Z) !
10Y
Substituting
n-Z k
(y_z)n-Z B(n-l)(z)dz+ L LB(k)(O+)
n
k=O k! n
into (3.16), and using the fact that (n-l)-divided differences of
polynomials of order n - 2 vanish, we obtain
n g(x) =
IT
1
(n-Z)!
roo [ Z
10
Z (0 _ z)n+-Z] Bn(n-l) (z)dz .
xl'···' xn;
Finally, setting
r > 0 ,
it follows from (3.6) and (3.14) that
oe
21
n
Since g and go are p.d. f. 's on lR ,
1 =
J
n
lR
g(x)dx =
~O dr • r- n fer) J n dx go(r- l x) = hO f(r)dr ,
lR
oo
so that f is a p.d.£. on [0,00) with d.f. F (appearing in (3.5)).
The
proof is now completed in a manner similar to that used in the proof of
o
Theorem 2.1.
The following property will be useful in the sequel.
The random vector X de fined by (3 .4) has independent
2
standard Cauchy components if and only if R is F-distributed with nand
TI-lEOREM 3. 2 •
n degrees of freedom, i.e., R has the p.d.f.
f (r) = 2r(n){r(n/2)}
-2
n
Proof.
r
n-l
2 -n
(1+r)
,
r
~
(3.17)
0 •
For the "if" part, it suffices to show that ¢leu)
equivalently that (RU
l
where B is beta-distributed with parameters n/2 and
2
!.:
2
or
)/115J. is standard Cauchy. Using 3.197 (4) of
2
Gradshteyn and Ryzhik (1980), one first shows that R /D
U (R /D )
l
l
= e -u,
is standard Cauchy.
~,
l
k B/(l- B) ,
and then that
The "only if" part follows from the
one-to-one correspondence between ¢l and F in (3.1), established in
Theorem 3.3, which uses only the "if" part of Theorem 3.2.
o
The following result provides an alternative characterization of the
class
<I>
n
(1), similar to that given in Theorem 2 of Cambanis, Huang and
Simons (1981) for
<I>
n
(2), which together with Bernstein's theorem (cf.,
Feller (1971), page 439) provides an effective way to check whether a
22
function
~
belongs to the class ¢ (1).
n
~ E
correspondence between
It also establishes that the
¢n (1) and the distribution function F in
(3.1) is one-to-one.
TIIEOREM 3.3.
~
E
¢n(l) (n
~
1) if and only if ~ is continuous and
bounded, and
s
~
0 ,
(3.18)
is the Laplace transform of a nonnegative random variable.
The
distribution function of this random variable is the distribution
function F appearing in (3.1).
Proof.
The "only if" part, as well as the claim concerning F,
follow innnediately from (3.1) and the "if" part of Theorem 3.2, which
implies that
s
~
0 .
Conversely, let (3.18) be the Laplace transform of a nonnegative random
variable whose distribution function is F, and define
the right side of (3.1).
condition H (s) ~
We will show that
fa h(sr) f n (r) dr = 0
continuous function h (in our case, h
~
=
~*
~*
E
¢ (1) using
n
by showing that the
for s ~ 0 occurs for a bounded
=~ -
~*)
only when h is identically
zero on [0,00). But for s > 0, using (3.17) and
-n
-1 roo n-1
x
= {r (n)} 10 v
exp ( -vx) dv, we obtain
.e
H(s) = 2{
r() 2
2
n-1
n } sn rno dv e -s v • v
r(n/2)
• u
n-l
10 du
h(u),
e- VU
2
(3.19)
23
which, in words, is a multiple of a Laplace transform (in s2) of a
multiple of a Laplace transform (in v) of the function (0) (n/2)-1 h(j;).
Because of the uniqueness property of Laplace transforms, it follows from
(3.19) that H(s)
=
0, s > 0, only if h
=
0 a.e.
Since h is continuous,
it must be identically zero.
4.
D
PROPERTIES OF l-Sm1ETRIC DISTRIBlITIONS
In this section, we study certain properties of random vectors
x=
(Xl"'" Xn ) with 1-synnnetric distributions. Analogous and more
extensive properties of 2-synnnetric distributions were established in
Cambanis, Huang and Simons (1981).
It is apparent from (3.4) that unless X = 0 UJith probability one
(corresponding to R
•
= 0),
the components of X have finite pth moments if
and only if -1 < P < 1 and ERP <
00.
Thus the components of X fail to
have finite first moments except in the degenerate case.
We now show that if
~o
random vectors have a joint I-symmetric
distribution, then the conditional distribution of one of them given the
The following lemma, whose proof is
straightforward and therefore omitted, will be needed. 2
other is also I-symmetric.
Let Rl and RZ be nonnegative random variables~ Zl and
Zz be random vectors~ X. = R.Z. (i = 1,Z)~ and assume the random vectors
LEMMA 4.1.
111
(Rl'RZ)' Zp and Z3 are independent.
Then the conditionaZ distribution
of Xl given Xz = X z is given by
.e
(4.1)
24
A regular conditional distribution of Xl given X2
=
x 2 can easily
be described, but for the sake of brevity it will be omitted.
In order to simplify the notation of the stochastic representation
(3.4), we will write
[~
THEOREM 4.1.
,..,~~J
=
Let X k R(U(n) /In(n)) ,.., Sn(l,<P) and X = (Xl,X Z),
where Xl is m-dimensional, 1 ~ m < n.
Then the conditional distribution
of Xl given X = x is given for almost every value of X2 by
2
2
(XlIXz=x z )
kR
X
where the distribution of R
X
z
U(m)
z ID(m)
(4. Z)
,
is defined implicitly by (4.4) below, and
the right side of (4.Z) is a stoahastia representation of the type
described in (3.4).
A regular conditional distribution of Xl given X =
2
defined which properly reflects (4.2) and (4.4).
X
z can be
The distribution
function F
x 2 of RXz can be expressed in tenus of the distribution
function F of R, but the resulting expression is too complicated to be
useful.
Thus (X1Ixz=x2) ,.., Sm(1,<P x ), where <PX is defined by (3.1)
2
z
with F replaced by F , and with <PO (in (3.1)) defined by (3.Z) with n
X
replaced by m.
.e
z
25
Proof of Theorem 4.1.
X = (X
X)
l' 2
[Rmn
=L R
S
The trick is to express X stochastically as
[
rr:mI'
IDl.m J
mn
2]~
l-R
rrm.
U(m)
u(n-m) ]
(4.3)
ID(n-m)
1- S2
mn
2 k
2-k
and to apply Lennna 4.1 with R = RR /S ,R = R(l- R ) 2 (l-S ) 2,
mn
mn
2
mn
mn
1
Z = U(m)/ID(m) and Z = U(n-m)/ID(n-m)
so that one has
1 2 '
R
x2
•
•
e
k
(RRmn
Smn
In (4.3) and (4.4),
I\nn'
2]~
I R 1 - Rhm
[
(4.4)
1-S 2
mn
R~
and
S~
are both Beta(m/2, (n-m)/2), and R,
Smn' u(m), u(n-m), D(m), and D(n-m) are jointly independent.
The
bases for (4.3) are the stochastic representations
of which the first is validated in Cambanis, Huang and Simons ((1981),
Lennna 1) and the second follows similarly from the fact that
1:
(n) _
D
2
2
o
- (D l ,···, Dn ) - (U l '···' Un) .
If X
= (Xl ,X 2)
'" Sn(l,<P) with <P(u)
= exp(-u),
then as noted
previously, the components of X are i. Ld. and standard Cauchy.
It
follows that the random vectors Xl and X , referred to in Theorem 4.1,
2
are independent, and thus the conditional distribution of Xl given X2
.e
does not depend upon X .
2
This motivates the following theorem, which is
a direct analogue of Theorem 6 in Cambanis, Huang and Simons (1981); it
has an identical proof which is thus omitted.
26
TI-IEOREM 4. Z•
If X
=
(Xl' X ) has a l-symmetroic distroibution.. then
2
the conditional distroibution of Xl given X
z
does not depend on X 7.Jith
z
probability one if and only if the components of X are i.i.d. scaled
Cauchy random varoiab les .
It
is unnecessary in Theorem 4.Z to say that "the components of X
are" both "i.i.d. and scaled Cauchy random variables".
Suppose X is
I-symmetric with characteristic function of the form ¢(Itll +... +. Itnl).
If X even has pairwise independent components, then it easily follows
that ¢ satisfies the functional relationship ¢(u+v) = ¢(u)¢(v) for
u,v
~
0, from which one can show that the components of X are (i.i.d.)
scaled Cauchy random variables.
Conversely, if the components of X are
scaled Cauchy random variables, then ¢(u) = exp( -cu) , u
c
~
~
0, for some
0, and it easily follows that they are i.i.d.
•
s.
BIVARIATE!.2 -SYMMETRIC DISTRIBlITIONS
We have found the class
Z(!.2) to be an enigma. For a long time, we
had what we thought was its primitive. Everything in the class is a
<P
I
scale mixture of this "pseudo-primitive".
apparent that, for certain members of
But it eventually became
Z(!.2) , the distribution function F
in (Z .1) must be replaced by a signed measure (still on [0,(0)). We
<P
interpret this as encouraging evidence that there does exist a true
primitive ¢o for
Z(!.2) , one for which (2.1) holds, with F a genuine
distribution function for each ¢ E <P (!.2) .
Z
.e
<P
27
Our pseudo-primitive is closely related to the primitive for
L
!::
-
~2(1);
-1:
it is stochastically representable as (XO,YO) = (U/(BlBi), V/(BlBi)) (cf.,
(2.6)), where B.1 = (I-B.)
(i = 1,2), (U,V) is lillifonnly distributed on
1
the lillit circle of JR2 , Bi ,.., Beta(~ ,~) (i = 1,2), and (U,V) , Bl , and B2
are independent. That (X O,Y0) has a characteristic flillction of the
right form is easily shown with two applications of Proposition 2.1.
Not only are we lillcertain about the existence of a primitive for
~2 (~),
but we are even lillcertain that (XO' yO) is "factorable", i. e. ,
that (XO' YO) k R(X!, Y1) for some (Xl' Y1) which has a characteristic
flillction of the right kind for
~2(~)'
and some R, independent of (Xl' Y1) ,
which is a nonconstant nonnegative random variable .
•
•
6.
n-DIMENSIONAL VERSIONS
In this section, we characterize the class of n-dimensional versions
of the symmetric one-dimensional stable laws and address some related
questions.
The connection between n-dimensional versions and a-symmetric
distributions is discussed in Section 1.
Recall that, according to Eaton (1981), the distribution of an
n-dimensional random vector X = (Xl"'" X ) is an n-dimensional version
n
of the distribution of a symmetric random variable Z if
tlXl + ..• + tnXn k c(t)Z for t = (t l , ... , t n) E lRn with cC'o ~ 0 for
all t. We prefer here to delete Eaton I s requirement that c(t) be
strictly positive for all nonzero t, which has the effect of excluding
X's with linearly dependent components.
We note in passing that c(t)
has to be a homogeneous function of degree one, i.e., c(st)
.e
for each scalar s and t
about its properties.
E
n
lR.
= Islc(t)
Beyond this, little is known, in general,
28
Eaton completely characterized the class of n-dimensional versions
2
when Z has a finite positive variance:
characteristic function of Z.
Let <I>(u) denote the
Then the n-dimensional versions are the
distributions of n-dimensional vectors AXO arising from kxn matrices A
of rank k, 1 ~ k ~ n, where Xo . . . Sk(2 ,<1». The function c(t) assumes
the form c(t) = (t L:t')
k
\
t
E
n
lR , where L: = AA'.
Thus, there exists an
n-dimensional version, with linearly independent components, associated
with such a Z if and only if <I>
E ~n(2).
Eaton showed, as well, that the assumption of finite variance is
critical.
In particular, he observed that when Z is a symmetric stable
random variable of index a, having characteristic function exp(-Iul a ),
•
the symmetric n-dimensional stable laws of index a are n-dimensional
versions.
When 0 < a < 2, Z has an infinite second moment, and the
class includes many examples which cannot arise from a linearly
transformed 2-symmetric Xo. The characteristic functions of symmetric
n-dimensional stable laws of index a assume the form
E exp{itX'}
= exp{-fltu' la l1(du)},
t
lRn ,
E
(6.1)
where 11 is a finite (symmetric) measure on the unit sphere in Rn
(u E IRn , uu' = 1). Thus
t
E
n
R
,
(6.2)
where
.e
c(t) = (fltu' la l1(du))l/a ,
t
E
n
lR
.
(6.3)
29
The following theorem shows that there are no other n-dimensional
versions.
TIIEOREM 6.1.
Let Z be a symmetric one-dimensional stable random
variable of index a.
The class of n-dimensional versions of Z coincides
with the class of symmetric n-dimensional stable laws of index a.
Proof.
Because of the previous discussion, it is sufficient to show
that a random vector X which satisfies (6.2) is symmetric n-dimensional
stable of index a.
will be enough to show that if Y is an independent
copy of X, then for all real r and s, rX + sY k (Irl a + Isla)l/a X.
Equivalently, we will show that (rX+sY)t' k (Irl a + Isla)l/a Xt',
•
t
E
It
n
:rn:.
But for real u,
Eeiu(rX+sY)t' = Ee iurXt' Ee iusYt'
= Ee iurc(t)Z
Ee iusc(t)Z
o
The foregoing suggests a (perhaps natural) generalization of the
a-symmetric distributions to those whose characteristic functions assume
the form
E exp{itX'} = ¢(fltu' la ~(du)) ,
with the attendant classes Sn (a, ~,¢) and
E;:rn:n ,
<P (a,~)
(obvious analogues of
n
plays the role of a scaling
n (a)) . The new parameter ~
parameter and corresponds to a covariance matrix l: when a = 2 via
f I tu' 1 2 ~ (du) = tLt'. Except for a few special cases, no information
Sn (a,¢) and
.e
t
<P
seems to be available on these distributions and classes.
30
ACKNOWLEDGEMENf
The authors wish to thank Professors Norman Jolmson and
Hari Mukerjee for helpful conversations pertaining to the contents of
this paper.
REFERENCES
[1]
BRETAGNOLLE, J., DACUNHA CASTELLE, D., AND KRIVlNE, J-L (1966).
Lois stables et espaces LP.
Ann. Inst. H. Poincare, Ser. B 2
231-259.
[2]
CAMBANIS, S., HUANG, S., AND SIKlNS, G. (1981).
elliptically contoured distributions.
[3]
EATON, M. (1981).
On the theory of
J. Multivar. Anal., to appear.
On the projections of isotropic distributions.
Ann. statist. 9 391-400.
[4]
FELLER, W. (1971).
An Introduction to Probability and Its
Applications, Vol. 2, 2nd ed.
[5]
GRADSHfEYN, 1.S., AND RYZHIK, 1.M. (1980).
series, and Products.
[6]
Wiley, New York.
Tables of Integrals,
Academic Press, New York.
ROBERTS, A.W., AND VARBERG, D.E. (1973).
Convex Functions.
Academic Press, New York.
[7]
SCHOENBERG, 1.J. (1938).
fmctions.
Metric spaces and completely monotone
Ann. Math. 39 811-841.
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1----------.----------.-----.--.-----.----.20. A eST R AC T
If fie
1.1
(Con'lnut~
t)f)
r6 ,.(OrSt> ..ddt'
Ct'" ~;iJf\' 'uhf
ide:)';
.__.
h)· bloc k f"J;'l:1'f'r)
._.
_
A random vector is said to have a I-symmetric distribution if its
characteristic function is of the form ¢(Itll
+ ..• +
Itnl)
I-symmetric
distributions are characterized through representations of the admissible
functions
~
and through stochastic representations of the random vectors, and
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