Journal of Mathematical Sciences, Vol. 92, No. 3, 1998
STRICTLY STABLE LAWS FOR MULTIVARIATE
RESIDUAL
LIFETIMES
UDC 519.2
A. A. B a l k e m a (Amsterdam, Holland) a n d Y o n g - C h e n g Qi (Peking, China)
The paper is organized as follows:
(1) A probabilistic part. We introduce multivariate residual lifetimes and discuss different ways of investigating their
asymptotic behavior. We briefly go into the relation with multivariate record sequences. We introduce the concepts
of stability and strict stability.
(2) An algebraic part. Our problem: to describe the possible connected subgroups of the group of coordinate affine
transformations on R a. We give a brief introduction to Lie algebras based on Z. J. Jurek and J. D. Mason. This
enables us to give a complete answer to the problem.
(3) The description of the strictly multivariate residual lifetime distributions.
Introduction
Residual lifetimes play an important role in renewal theory, survival analysis, and queueing processes. It is known
that the limit laws for residual lifetimes are related to the extreme value limit laws. In this paper, we shall investigate
the limit laws in the multivariate setting. The paper is organized as follows:
(1) A probabilistic part. We introduce multivariate residual lifetimes and discuss different ways of investigating their
asymptotic behavior. We briefly go into the relation with multivariate record sequences. We introduce the concepts of
stability and strict stability.
(2) An algebraic part. Our problem: to describe the possible connected subgroups of the group of coordinate affine
transformations on R d. We give a brief introduction to Lie algebras based on Jurek and Mason [2]. This enables us to
give a complete answer to the problem.
(3) The description of the strictly multivariate residual lifetime distributions. We may restrict attention to ddimensional subgroups ~ which do not factorize and do not contain a translation along a vector in [0, oo) d. In appropriate
coordinates such a group G consists of all transformations 7 on R e of the form
3,(s
t GR,
b e L,
where L is the hyperplane {~ = 0} with ~(s = xt + --- + xd. Assume that d > 1. The noncomposite, strictly stable,
residual lifetime distributions on [0, oo) d properly normalized have densities of the form f~ or g~ with a > 0 and
L(~)=
~(~+1)'"(~+d-1)
(l + ~),'+~-,
'
g~'(~)= ~(o,+ l)...(~ +d-
I)(I - ~);-'
with ~ = xl + "'" + Zd as above, or are uniformly distributed over the d - 1-dimensional simplex generated by the d
vectors e'l,- .-, gd of the standard basis of R d
1. M u l t i v a r i a t e R e s i d u a l L i f e t i m e s
For a random vector X in R d, we define the residual lifetime X : of X in the point/Y as the vector X -/:7 conditional
on X >_/Y. In this setting, it is convenient to work with the tail function R of X defined by
R(s
= P{X _ s
= P{X e [~,~)}.
Then the residual lifetime X : is well defined for any point iff, where the tail function is positive and X : has tail function
a:(x) = R(:+ ~)IR(p-3, i >_ 6.
This relation determines a probability distribution on [0, oo) d.
Ry(y-) = R:(ff V O) outside [0, e~) d.
We extend R: to the whole space R a by setting
Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajd6szoboszl6, Hungary, 1997, Part I.
1072-3374/98/9203-3873520.00 9
Plenum Publishing Corporation
3873
Note that s ~-* R ( - ~ ) is the distribution function (d.f.) of - X . One can express the tail function in terms of the
d.f. of X. However, for d > 1 this expression is rather unwieldy.
What can one say about the asymptotic behavior of the residual lifetimes X~ as/7 increases and R(p-') tends to zero'?
Take a point ~'on the upper boundary of {R > 0}. Note that ( m a y be finite or infinite. We assume that R(/7~) - - (}
for fin ]" ~. Do there exist normalizations ")',~so that the sequence of random vectors Z,~ = %~(X~,) converges in law to
a nondegenerate limit vector? If so, what are the possible limit distributions?
1.1. C o o r d i n a t e a t f i n e t r a n s f o r m a t i o n s . Before we can answer these questions we have to decide on the group of
normalizing transformations. Since these have to preserve the order on R d, we choose coordinate afline transformatio~Js,
cats for short. These have the form
~'(~') • ('~1(271),-
..,"[d(Xd)),
2~-~-- (27 1 .....
27d),
where 7i(u) = ea'u + bi for i = 1 , . . . , d is a positive affine transformation on R.
It is well known that the positive affine transformations eau + b, a,b E R, on R form a group 7). This group
noncommutative since, for instance, 2(u + 1) # 2u + 1. Cats form the group 7~a. We shall also consider subgroups
pd. Well known are the d-dimensional group of all translations ~ ~ ~ + b, b E R a, and the one-dimensional group
multiplications s ~ etZ, t E R. In the second part of this paper, we shall describe all connected subgroups of 796.
A random vector Z has a nondegenerate distribution if each of the components is not a.s. constant. If X
nondegenerate, then c~(X) and fl(X) have the same distribution for c~, fl E 79a if and only if ~ = ft.
is
of
of
is
1.2. S t r i c t s t a b i l i t y . A second, more difficult question which we have to settle is what limit sequences we shall
consider. Let (/~) be a given increasing sequence. For simplicity, assume that the sequence is strictly increasing,
/Yn+t E (/~n,oo). Then it is not difficult to see that for any vector Z with a nondegenerate distribution we can construct
a vector X such that the residual lifetimes Xr properly normalized converge in distribution to Z. The class of limit
distributions is the set of all nondegenerate probability distributions.
The situation becomes more interesting if we consider the asymptotic behavior of Xr
for a strictly increasing
continuous curve /~(t), t >_ 0, such that R(ff(t)) ---, 0 for t --* oo. In that case, one can use arguments well known
from extreme value theory to conclude that the limit vector Z has to satisfy certain stability conditions. Under weak
regularity conditions there exists a one-parameter group of cats 7 t and a curve ~ t ) , t > 0, so that Zr
= 7~(Z) for
t _> 0. This still leaves us with a very large class of limit distributions which depend on the b o u n d a r y point ~" and on
the precise form of the increasing curve q-(t) leading to this boundary point.
Now take a simple example like the uniform distribution on the unit disk in R 2. Take a point ~'on the boundary
in the positive quadrant (0,oo) 2. Then it is clear that the residual lifetime limit exists for all increasing sequences
~', ]" ~'. The limit is the uniform distribution over the t r i a n g l e / k formed by the origin and tile two coordinate vectors in
R 2. It is also clear that in contrast to multivariate extreme value theory the limit depends only on the local behavior
of the distribution of X around the boundary point ~'. Our limit distribution now is not stable for one particular
one-dimensional subgroup of cats, but for a two-dimensional subgroup. This subgroup ~ o f p 2 is so large that it can
map the origin into any interior point of the triangle.
The Lebesgue measure on the half plane H = {x + y < i} is semi-invariant under a group of cats. This group
G C p2 consists of translations parallel to the line {x + y = 1} and multiplications from a center on this line. The
set ~({~) of vectors 7(0), 7 E ~, fills the half plane g . In particular, it covers the triangle /k = [0, oo)2 N H. For each
/~ E A , there is a unique 7 E ~ which maps 1) onto/~. The cat 7 transforms r into a p r o b a b i l i t y measure 7 ( r ) , the
uniform probability measure o n / k . The measure 7 ( r ) is a multiple of the restriction of r to the t r i a n g l e / k .
D e f i n i t i o n . A probability measure r on [0, oo) d is strictly stable for residual lifetimes if there is a connected group
of cats, ~ C 7)d, so that ~(6) contains a nonempty open subset of [0, oo) d, and so that 7(7r) is a multiple of the
restriction of 7r to [7(6), oo) for each 7 E G with 7(()) >_ 0.
1.3. S t a b i l i t y f o r r e s i d u a l l i f e t i m e s a n d f o r r e c o r d s . Let X be a random vector with probability measure 7r
and tail function R. It is natural to call 7r stable for residual lifetimes if for each ig for which R(p-) is positive there
exists a cat ~, such that X~ is distributed like 7(X). We shall not go into the general theory of stable residual lifetimes
here. We only mention
PR.OPOSITION 1.1. Strictly stable residual lifetimes are stable.
P r o o f . Let X be strictly stable. Then X _> 0. We may assume that ff > I). Indeed, set ~" = ffv 1). Then X~ + ff and
X4-+ ~'have the same distribution. From the structure of the connected subgroups of T~d below it will follow that G({))
contains the set {R > 0} fq [0, oo) d. Hence the condition for stability is satisfied.
In a sequence of vectors (Z-,), an element s is a record if s > xi for all i < k. Obviously Zh is a record, but a
sequence need have no other records even if s ~ oo coordinatewise. However, it will then contain a subsequence of
records. The sequence fin = xk, is such a record sequence if kl < k2 < . . . and ffl _< if2 _< . . . .
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Now consider a sequence (X~) of independent vectors from a common distribution 7r. Conditional on X1 = 13".tbe
first (random) index K > 1 for which XK >_ iYyields a random vector such that Xh- >_ i6 a.s. Assume that R(I~ > O.
Then the vector X K -- ~ is distributed like the residual lifetime XrT. This leads to a slightly different concept of
stability, where we consider the tail function R ( ~ ) / R ( p - ) on Lff,oo) and restrict it7 with R(p-) > 0 to lie in the support of
the underlying probability measure rr.
E x a m p l e . For any centered lattice L C R d and any vector ff E (0, oo) a, the measure with atoms of weight e -(a'e)
in the point aS of L is finite on [0, c~) d and determines a probability measure on [0, cxD)d with tail function R which
satisfies the stability relation
n ( ~ + i.)ln(p-" ) = / ~ ( ~ ) ,
as E L n [0, oo) ~.
This probability measure is not strictly stable for residual lifetimes, and it is stable for residual lifetimes only if L is a
product of one-dimensional lattices in different coordinates.
Strict stability implies a certain homogeneity of the measure: the measure to the Northeast of any point/7 in [0, oo) d
is the same as the original measure up to a multiplication and some scalings. This is also true for stability of conditional
records, but does not hold for stability of residual lifetimes. Let U be a s t a n d a r d exponential random variable (r.v.) on
[0, oo). Then [U] has a geometrical distribution and is stable, but lacks the homogeneity of the exponential distributions,
so too the vector (X, X2), where X -- e v.
2. C a t s
Coordinate affine transformations, cats for short, form a group of simple transformations acting oil the vector space
R d. Since they preserve order, cats are convenient normalizations for multivariate extremes and residual lifetimes. A
cat 7 has the form
7(as) = (7~(~1),...,7d(~)),
~=
(~,,...,~d)
E
RL
where the 7i(u) = ea'u + bi are positive affine transformations on the real line.
This paper is concerned with the following question: describe the subgroups of the 2d-dimensional group 7~a of cats.
We restrict ourselves to connected subgroups. In particular, we are interested in d-dimensional subgroups Ca such that
the images T(P-'), 7 E Ca, of a given point i~ in R a form an open subset of g a. Our aim is to identify probability measures
on g d which are strictly stable for residual lifetimes. This is related to the problem of finding measures/~ on g d which
are semi-invariant under all cats 7 in the group Ca: the image measure 7(/~) is a constant multiple of/~ with the constant
depending on 7The group 79 of positive affine transformations eau 4- b on the real line is noncommutative, and so is the group 7~
of cats. This means that Lie algebras form the appropriate tool for our investigations. Since this paper is intended
for probabilists rather than algebraists, we shall give a brief introduction to the theory of Lie algebras, basing our
arguments on concepts like semigroup and infinitesimal generator, which are well known from the theory of Markov
chains, and using only the matrix theory for multivariate Gaussian distributions.
This part consists of four sections. We first give some illustrative examples, then define the Lie algebra for subgroups
of the group of invertible matrices, show how the group -pd fits into this framework, and finally give an explicit
description of the connected subgroups of-pd.
2.1. E x a m p l e s . This section introduces a number of concrete subgroups of the group "Pd. The vector space R d is
a commutative additive group. The connected subgroups are linear subspaces. For noncommutative groups it is tess
simple to determine the subgroups.
E x a m p l e 1. T h e o n e - d i m e n s i o n a l s u b g r o u p s . In the group 7~ of positive afl]ne transformations on the real
line, each element 7(u) -- eau 4- b generates a one-dimensional subgroup. If a = 0, this is the group of translations
7~u = u 4- bt, t E R; otherwise, it is the group of multiplications 7tu = eat(u - c) 4- c with center c = b/( l - e~). The
same holds in p d . Let 7 be the cat 7 = ( 7 ~ , . - - , 7d) with 7, E P . Then 7' = ( 7 { , - - . , 7~) is a one-dimensional subgroup
of -pa.
We restrict our attention to subgroups Ca of T'~ with the property
7 E Ca :::=:~ 7t E Ca,
tER.
Write D = { l , . . . , d } . A nonempty subset E C D and any subgroup Ca of T'D determine a subgroup CaE o f ' P c by
restricting the action to points in R E. This projection is defined by 7E(as) = (Ti(xi))i~E for as E R e. In particular, ~i
is a subgroup of 7' for each index i. One has the following simple result.
PROPOSITION 2.1. Let Dl , 9 .., D,,~ be a partition o f D into n o n e m p t y subsets. A n y s u b g r o u p Ca o f ' P d is a s u b g r o u p
o f the p r o d u c t g r o u p CaD, x 9 . 9 x CAD,. C T 'a.
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E x a m p l e 2. T i l e c o m m u t a t i v e s u b g r o u p s . For each i = 1 , . . . , d, choose an element 7/ E T), not the identity.
Then every vector t = (tl . . . . , td) determines a nnique cat ~,t = ( 7 1 ' , - - - , 7 J ) and the map t ~-* 3" is an isomorphism of
the additive group R d onto the subgroup jC of cats 7 t. Linear subspaces of R d correspond to closed connected subgroups
of ~. These subgroups all are commutative. On the other hand, any commutative subgroup 7/ of'Pal has this form.
Indeed, let "Hi be the corresponding subgroups of "P. These groups "Hi are commutative and hence are contained in a
one-dimensional subgroup ")'~, s E I'C, of ~P. Now Proposition 2.1 states that 7-( is contained in tile commutative group
~, x...x'Ha.
E x a m p l e 3. A n o n c o m m u t a t i v e s u b g r o u p . The set of all cats of the form 7(x) = e t x + b" with t E R and
b" E R d is a noncommutative subgroup A C T'd. It contains the group of all translations and the group of all
multiplications by a constant. For any linear subspace L of R d, the set of cats 7(s = e'Z" + b, b E L, is a subgroup
A(L) of A.
There are other connected subgroups in .A.
Let ? b e any vector in R d. Then A contains the one-dimensional subgroup of all multiplications et(s- - c-) + 5" with
center 5". We can add a linear space L of translations, thus obtaining a group with elements e t Z q - b, b E L~, where Lt
is the affine manifold parallel to L passing through the point (1 - d)~.
In order to avoid these clumsy subgroups, we introduce the convention that we choose coordinates on the space R d
so that the origin becomes tv.e center of the multiplication whenever this is practical. Note that R ~ is an affine space.
We are free to choose the origin and scale in each coordinate.
With this convention, commutative subgroups are determined by a linear subspace L of R d and a subset Do C D.
~l
By definition, C(Do, L) is the set of cats 3' = (c~L
. . . . , a dtg ) with t = ( 1 ~ , . . . , td) E L and a~(u) = u + 1 for i E Do and
oi(u) = eu elsewhere.
E x a m p l e 4. Consider the group ~ C 792 generated by the two elements a ( x , y) = (2x, y + i) and fl(z, y) = ( x + 1,2y).
The four-dimensional group 7 '2 is transversed by two systems of curves, c~t7, t E R, 7 E p 2 , and fit"/. Think of these
as two railway systems. The group G consists of all points in 7 '2 which can be reached by using those two systems
starting from the identity element. We claim that jC = T'd.
First note that a/3 # fla and that 7- = (fla)-l(afl) is a translation - - the multiplicative factors cancel. Indeed,
straightforward calculation shows that 7-(z, y) = (x + 1/2, y - 1/2). One can prove that j(j also contains the translations
7-~, t E R. Now c~-~r2c~ again is a translation, o'(x,y) = (x + 1/2, y - 1). Since 7- and (r are linearly independent, we
conclude that the group ~ contains all translations. Then it also contains the multiplications (2x, y) and (x, 2y). This
allows us to reach any point 7 in P : .
The last example shows that for the noncommutative group T~2 the problem of determining the connected subgroups
is of a different nature that of determining the linear subspaces of R 4. The group T~d is less homogeneous than the
additive group R ~d and the subgroup generated by a set of k elements may have dimension > k . The arguments in
Example 4 can be used to analyze the group ~ generated by any finite number of cats in ~ a . However, it is simpler
to work with Lie algebras. This means that one works with the infinitesimal generators rather than the one-parameter
subgroups.
2.2. T h e g r o u p G L ( d ) a n d i t s L i e a l g e b r a . This section treats the group GL(d) of invertible d • d matrices. We
introduce the exponential function and the logarithm and define the Lie algebras associated with connected subgroups.
The arguments should be understandable to anyone acquainted with semigroups of(finite) Markov matrices p t , t > 0,
and their infinitesimal generators A = l i m t ~ 0 ( P ~ - l)/t. In fact, since all our matrices are finite dimensional, there are
no convergence problems and the theory is rather simple.
A matrix P is invertible if its determinant does not vanish. The set GL(d) of invertible d • d matrices is the open
set, {det 7~ 0}, in the d2-dimensional space A4(d) of all d • d matrices. The set GL(d) is a group with identity [, the
identity matrix. W i t h each connected subgroup G of GL(d) we associate a linear subspace A(jC) of.hA(d). Geometrically
it is the tangent space t o ' t h e group G in the point I. It consists of the infinitesimal generators of all one-parameter
subgroups p t , t E R, of ~. We shall follow [2] and define the Lie space A(jC) of the subgroup jC as the set of all matrices
A for which there exists a sequence (P~) in ~ tending to I and a sequence of positive constants e ~ 0 with e ~ + l / e , ---* 1
so that
(Pn
-
I)/er~ ~
A.
In a finite-dimensional space, all convergence concepts coincide. However, for our definition of the exponential
function and the logarithm we need a norm on the set .h4(d). Let [1-[1 be any norm on R d. Now define the operator
norm on .M(d) in the usual way by HAll = sup [[As where we take the sup over all vectors s with norm 1[~1[ = 1. It
is well known that [ l 11 is a norm on the linear space .M(d) with the additional properties [ l l l [ = 1, [[ABI[ < [[A[I[tB[[.
We can now define the functions exp(A), ( I - A ) - t , and log(1 + A ) in terms of power series. Since [[A"[[ < [Id[] '~, the
power series of exp(A) converges for all m E .M(d) and the power series for (1 - A ) - ' and Iog(l + A) converge for all A
3876
with norm IIAII < 1. Note that exp(A + B ) = e x p ( A ) e x p ( B ) if A and B commute. In particular e x p ( - A ) e x p ( A ) = I,
and hence exp maps .A4(d) into GL(d). For each A # 0 9 Ad(d), the map t ---, exp(tA) is a homomorphism of the
additive group R to the one-dimensional subgroup exp(t.A), t 9 R, because
e x p ( l A ) e x p ( s A ) = exp((t + s)A),
s,t 9 R.
Conversely any one-dimensional subgroup p t , t 9 R, of GL(d) has this form. Tile matrix A is the infinitesimal
generator of the group p t . This follows from the fact that [IPt - 111 < 1 for small enough t, which allows us to take
the logarithm. Note that exp(log(A)) = A for IIA - III < t and log(exp(A)) = A for e IIAII < 2 ( b y comparison of the
power series). Thus, there is an open ball U centered at I which is mapped by exp onto an open neighborhood V of 0
in A4(d). This map is a bijection with inverse log: V ---, U.
We shall now prove a number of simple results about the spaces A(Ca).
PROPOSITION 2.2. A(aC) is a linear space.
P r o o f . If (P~ - I)/e,~ ~ A, then (P, - I)/ee ~ A / c for any c 9 (0, oo). If also ( Q , - l)/rl,~ ~ B, then we may
assume that e , = r}~ by taking appropriate subsequences and find
(Q.-I]
P~Q.-l_p~
Cn
\
+ P n- -- I
Cn
J
,B+A
en
since Pn -~ I in the first term. Similarly, (p,~l _ l ) / e , - - - A .
PROPOSITION 2.3. A 9 A(Ca) ~ exp(A) 9 jC.
P r o o f . Let (P,~ - [)/e,~ ~ A and set k , = [c/e] with c > 0. Then k,(Pn - I) ~ cA. For sufficiently small c, the
Poisson limit to the binomial distribution gives
P~" = ( l + _~,~)k. ----, exp(cA).
Not every linear subspace L of A is the Lie space of a subgroup jC of GL(d). This crucial observation follows from
the fact that the Lie space of a subgroup Ca has to be a Lie algebra:
A , B E A(~)==:::~[A.B]:= A B - B A 9 A(Ca).
(1)
The proof of (1). Let Pn = exp(A/n) and Qn = exp(B/n). Then P~Q,,PZ1Q~ l E Ca and
n2(PnQ,~P~XQ~ ' - I) --~ [A, B],
as is seen by writing out the product
( A
I+--+
n
A2
~
) ( B B '
I+--+
+
"'"
,,
)(
+
~
l---
A A 2 ) (
+
+
BA2
I---+
+
)
-
"'"
up to terms of order of 1/n 2.
THEOREM 1. Let A(Ca) be the Lie space of the connected subgroup Ca of GL(d). Then (1) holds.
R e m a r k . Conversely, one can prove that any subspace of the Lie space A(d) of GL(d) which is closed under the
Lie multiplication [-,-] in (1) is the Lie algebra of a connected subgroup of GL(d). We shall not give the proof here
since for cats this follows from the construction below.
2.3. C a t s a n d m a t r i c e s . A positive affine transformation u ---, e~u + b on R can be represented by a 2 x 2 matrix
P with top row (1,0) acting on column vectors (1, u)' in R 2 as follows:
P['=
,0) (1)=
(b
e"
Now consider the one-parameter subgroup e'=t(u - c) + c and take the limit ( P ' - t ) / t for t ~ 0:
,(,_,
g
C -- e a t c
0) ,(0o)
Cat -- 1
--Ctl
a
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The corresponding element of the Lie algebra is a matrix with a top row consisting of zeros. The b o t t o m row is
(-ca, a), where a is the exponent of the multiplicative part of the transformation and c the center. If we start with a
group of translations u + bt, t E R, we obtain a similar matrix with bottom row (b, 0). The Lie algebra corresponding
to the group P is two dimensional and can be represented by the row vectors (s, S), where ,5' is the exponent of the
multiplication and - s i s the center. If S = 0, then s is the translation. This gives the simple representation by row
vectors in R 2
c) + c
, ( - c a , a),
(2a)
u + b - - ~ (b, 0).
(21))
Cats can be embedded in GL(2d) as blocked diagonal matrices. A more economical representation is in G L ( I + d)
by matrices whose top row is ( i , 0 , . . . , 0). The cat s ---* (e'~'xl + b l , . . . , e~'~:Cd+ bd) then corresponds to the matrix
map
1
[1
0
...
2:1
I bl
e a~ ...
9
i
0
xa
b
0
".
.
.
.
0
Cant
I :dl)
/ 1)
eatxl +
bl
\ e a~Xd + bd
Below a top row (1, 0 . . . . ,0) are a translation vector b and a diagonal matrix with elements e ~ . The element of the Lie
algebra corresponding to the one-parameter group 3't generated by 3' is a matrix with top row zero, diagonal matrix
with elements ai, and column vector with elements -ciai if 3'i is a multiplication with center ci and bi if 3'i is the
translation xi + bi. The Lie algebra A(T'a) is 2d-dimensional and its elements may be represented by bivectors of the
form (g, S), where the second component S is interpreted as a diagonal matrix. The Lie multiplication has the simple
form
[(g', S), (~', R)] = ( S F - Rg', 0).
(3)
For cats, the exponential function is a bijection from the Lie algebra A(~Od) to the group p d . In fact, this map is a
homeomorphism but not a group isomorphism; the multiplicative property of the exponential function only holds for
elements whose Lie product vanishes. The inverse map, the logarithm, maps T'd onto A(T'a) and is given by (2) above.
We shall now describe the class of all subspaces of A ( P d) which are closed with respect to the Lie multiplication (3)
and the corresponding subgroups of-pal.
2.4. T h e s u b g r o u p s o f p a . We start with an explicit description of the subgroups of 79d. For ease of notation,
we choose coordinates on R d so that the center of the multiplication is the origin.
Let D 0 , . - . , D m with m > 0 be a partition of the set D = { 1 , . . . , d } , where we allow the set Do to be empty. For
k = 1 , . . . , m, let Lk be a linear subspace of R Dk, and let L be a linear subspace of R D~ x R m. The subspaces Li are
arbitrary. The subspace L has to satisfy two conditions, which will be specified below. Corresponding to this partition,
we write vectors s in 1%d as
:~ : (T.0,---,~'m), ~k ~ p Dk.
With the partition ( D o , . - . , Din) and the linear subspaces L, L 1 , . . . , L,n we associate a subgroup G o f ' P d . The group
consist of all cats 3' of the form
= (t0 + b-0, e',z
+
e " Z m + gin)
with
(b0,tl,...,tm) EL,
bk E L k ,
k= l,...,m.
First observe that G is a group since it is closed for multiplication and inversion.
The projections GDk are of the form A(Lk) C pVk for k >__ i and ~Do is a group of translations. If L = L0 • R m
for some linear subspace L0 of R ~176then G = GOo • "'" x G o , - If the subspaces Lk are trivial, L~ = {0} for each
k = 1 , . . . , m, then ~7 is the commutative group C(Do, L).
We shall now specify the two restrictions on the linear space L C R o~ x R " :
(1) For each k = 1 , . . . , m , there is a vector ff = ( f f 0 , u x , . . . , u m ) E L with u~ # 0.
(2) For each pair of indices 1 _< j < k < m, there exists a vector ff in L with uj # uk.
It is clear that these two conditions ensure that the partition ( D o , . . . , Din) is unique. If condition (I) is violated
for some index k _> 1, then ~Dk is a set of translations and we may add the set De to Do. If condition (2) is violated
for (j, k), we may merge Dj and Dk into one set.
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We claim that every connected group of cats has the form above provided we choose the origin in R a appropriately.
Equivalently, every linear subspace of A(7 )d) which is closed for the Lie multiplication (3) generates a group jC of tile
form above.
P r o o f . Let F be a linear subspace of A = A(T'a). We regard A as the space of all fimctions on {0, l } • D. Define
Fl by restricting the functions in F to {1} x D. Alternatively, Fl is the set of diagonal matrices A such that. (if, A) lies
in F for some vector ff E R d.
Define Do as the set of all indices i E D for which Ai vanishes for all A E F1, and call two indices i and j equivalent
if Ai = Aj for all A E FI. Let D a , , . . , Dm be tile equivalence classes in D \ Do. As fimctions on {1} • D, each element
of FI is constant on each of the sets {1} x Dk, k = 0 , . . . , m, and these subsets Dk are maximal.
We claim that the space F1 contains an element B with the property that /4i 7~ 0 for i E D \ Do and Bi r /?j
if i and j belong to different sets of the partition. Such an element can be constructed inductively. If the matrix
B (k) does not vanish on the sets D 1 , . . . , Dk and k < m, then choose a matrix A which does not vanish on D~+l and
define B (k+l) = B (~) + cA with c > 0 so small that the new B (k+l) does not vanish on any of the sets D l , . . . , Dk. A
similar construction can be used to ensure that Bi r Bj if i and j are not equivalent. (The line b'+ c5 will intersect a
hyperplane {xi = 0} of {x = xj } in at most one point unless both (7 and b" belong to the hyperplane.)
Let (b, B) be any element of F whose multiplicative component B has distinct nonzero values on the different sets
Dr, 9 Din. By an appropriate choice of the origin in R d we may assume that bi vanishes for i E D \ Do. lndeed, let F
be the vector with components ci = - b i / B i for i E D \ Do and ci = 0 for i E Do. The element 7 = exp(b, B) E P a has
the form ( T t , . . . , 7a), where ")'i is a translation (over bi) for i E Do and a multiplication (by e B') with center ci = - b i / B i
for i E D \ Do. Change coordinates by putting the origin in (~ E R a. Then "[i(aZi) ---- e B - i x i for i E D \ Do and the
corresponding element in the Lie algebra has the form (b, B), where Bb = (~.
The Lie product [(g, t3), (~, X)] of (g, B) with any element (g', X) of the Lie algebra F is the translation (BE, 0).
Repeating this multiplication, we see that r contains the translations (B'~2-, 0) for n = 1 , 2 , . . . and by linearity the
translations (P(B)E, 0) for any polynomial P. Let B(k) be the value of B on the set Dk. Since the values B ( 1 ) , . . . , B(rn)
are distinct and nonzero, there are polynomials Pk such that Pk(B) has the value 1 on Dk and vanishes off Dk. This
has the following consequence:
LEMMA 1. Suppose that ( s
is an element of F. Then so is (s
components zi~ agree with xi for i E D~ and vanish for i E D \ D~.
where 2.~ is the vector in R a whose
It follows that we can write F as the direct sum of m + 1 linear spaces F = A* + At + . - . + Am, where A~ is the
space of those elements of F which vanish off {0} x D~ and A* consists of the elements which vanish off E, where
E C {0,1} x D i s t h e s e t of indices (0, i) w i t h i E Do and ( l , j ) w i t h j E D \ D 0 . It is clear that A~ corresponds to
a linear suhspace L~ of R D~ and A* to a linear subspace L of R D~ x R.m. The subspaces L~ are arbitrary but the
vectors (g', s t , . . . , s,n) of L satisfy the two conditions (1) and (2) above.
We can now formulate the main result of the algebraic part of the paper.
THEOREM 2. Let ~ be a connected subgroup of the group "Pa. We can choose coordinates on the space R a so that
the group ~ has the form described above.
3. T i l e S t r i c t l y S t a b l e D i s t r i b u t i o n s
We begin with a description of a class on d-dimensional subgroups of p d
PROPOSITION 3.1. Let G be a connected subgroup o f P a o f dimension >_d which does not factorize and which does
not contain a translation along any o f the coordinate axes. Then we can choose coordinates in I:Ld so that ~ =- A ( L),
see Example 3, where L is the hyperplane {~ = 0} with ~(s = z~ + . . . -t- xa.
P r o o f . If the linear subspace L in the representation of the group jC has dimension do + m, where do = [Do[, then
factorizes: ~ = ~Do X - - - • ~ D ~ Now suppose that L is a nontriviM subspace of R D~ • R m and ~ has dimension
>_d. Then at least one of the m groups jCD~, k >_ l, has dimension [Dk[ + 1, and hence contains all translations in R D~ .
The following result is well known.
LEMMA 2. Let p be a finite measure on an open set U in the product L x M o f two vector spaces. Let V be the
projection of U on L. Suppose that tz is invariant for translations along M in the sense that # ( E + (6, y-')) = # ( E )
for any g E M and any Borel set E C U M U - (l?, y-'). Then there is a locally f/uite measure p on V so that # is the
restriction to U o f the product measure p • A on V x M , where A is the Lebesgue measure on M .
Now let # be a Radon measure on R a which is semi-invariant under the group jC: for each cat 7 E ~: there exists
a positive constant C(7) such that 7(#) -- C(7)#, where 7(#) is the image of the measure # under the m a p 3'- Since
7((;#) = CT(tO, we find C(c~fl) = C(~)C(fl). Thus C : jC ---. (0, oo) is a homomorphism to the multiplicative group of
reals. It follows that (?(c~[]c~- 1r 1) = 1 for all ~, fl E jC and hence for all translations 7"(2-) = ~ + t ' w i t h {'E L1 +" - -+ Lm.
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If G contains a translation of the form s
e'i, where gi is one of the d basis vectors in the s t a n d a r d base of R d, then
the measure i L factorizes by Lebesgue measure on the ith c o m p o n e n t . In particular, the measure /~ c a n n o t be finit.e
on any orthant [if, or of the space R a. This proves t h a t the t r a n s f o r m a t i o n group jC associated with a s t r i c t l y stable
probability measure 7r on [0, cx~)d which does not factorize has the form _(;/= A ( L ) of the p r o p o s i t i o n above.
THEOREM 3. Let r be a probability measure on the orthant [0, oo) o f R d. Suppose that for each ff in some
open set U C [(),oo) there is a cat 7 = 7(P-') such that 7(P-') = 0 and 7(7r) restricted to [0, oo) is a multiple, o f t . Let
R : [0, eo) a --* [0, 1] be the tail function o f 7r, R(s = 7r[:?,oo). If 7r is not a product measure, then up to a scale'
transformation o f the axes R has the form
R(E) = l / ( l + x l
+."+xd)
a,
R(:~) = (1-(2:1--~------l-Xd)) d+/3,
(t > 0,
/3> --l.
I f d = l, then R ( z ) = e - z is also possible, and for d > 1 we m a y take fl = d - 1 in the second relation above.
P r o o f . It is well known that for d = 1 the d i s t r i b u t i o n tails s t a b l e for residual lifetimes are of the form e - ~ , (l + x ) ~.
and ( 1 - x)~+ on [0, (>D) with a and fl positive (see [1]). For d > 1, there exists a subgroup G o f t 'a such t h a t the measure
r is semi-invariant under the action of G. Hence G = A ( L ) by the proposition above, under a p p r o p r i a t e coordinates
on R ~. The l e m m a above implies t h a t the tail function R is a function of the sum ~ = xl + -- - + zd. Invariance under
multiplication implies t h a t the tail function has the form C~ - ~ on [1, c~) d or C~c - ~ on [ - 1 , oo) d, where a and fl are
determined by integrability conditions. A change of coordinates then gives the desired result.
REFERENCES
1. A. A. Balkema and L. de Haan, "Residual life times at great age," Ann. Probab., 2, 792-804 (1974).
2. Z. J. Jurek and J. D. Mason, Operator-Limit Distributions in Probability Theory, Wiley, New York (1993).
3. A. W. Marshall and I. Olkin, "A m u l t i v a r i a t e exponential distribution," J A S A , 62, 3 0 - 4 4 (1967).
Department of Mathematics,
University of Amsterdam,
pl. Muidergracht, 24, 1018 T V Amsterdam,
Holland
e-mail: guus@wins, uva. nl
Department of Probability and Slatistics,
Peking University,
Beijing 100871, P R Peking,
China
e-mail: [email protected], cn
3880