ELSEVIER Journal of Statistical Planning and

Inference 63 (1997) 187 201

journal of statistical planning and inference

Characterization of multivariate distributions through a functional equation of their characteristic functions

A r j u n K. G u p t a a'*, T r u c T. N g u y e n ~ l , W e i - B i n Z e n g b a Department ~[' Mathematics and Statistics, Bowlim] Green State University, Bowlin~! Grcett,

OH 43403, USA b Department q/' Mathematics, Unil,ersity of Louisville. Louisville, KY 40292, USA

Received 7 April 1995; received in revised form 9 August 1996

Abstract

The multivariate distributions whose characteristic functions satisfy a given integrated func- tional equation are proved to be essentially multivariate stable (semi-stable) distributions. This generalizes the characterization of univariate distributions in Ramachandran and Rao (1970), Shimizu (1968, 1978), Davies and Shimizu (1976), and Ramachandran et al. (1988). Related characterization problems such as identically distributed, and zero regressions of linear statistics are also discussed. @ 1997 Elsevier Science B.V.

A M S class![~cation: Primary 62H05; Secondary 60E07

Keywords: Integrated Cauchy functional equation; Characteristic function; Infinitely divisible: Multivariate stable distribution; Semi-stable; Linear statistics; Regression

1. Introduction

Characterization of probabili ty distributions through properties of linear statistics

stems from characterization of the normal laws. Starting with the Kagan-Linn ik -Rao

theorem (Kagan et al., 1965) in 1965, Rao (1967) proved in 1967 that if Xi and

~ are independent and identically distributed random variables with E[XI] = 0 and

0 < var(X1 ) < vc, then the regression equation

E[alXi +a2X21blXl + b 2 X 2 ] = O , a.e. (1)

implies the normality of X1. The extension of the regression equation (1) from the

case of two to more random variables,

* Corresponding author. 1 Supported in part by NSF Grant OSR-9452895~ and by a grant from the University of Louisville.

0378-3758/'97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PIIS0378-3758(97)00010-4

188 A.K. Gupta et aL /Journal of Statistical Planniny and InJerence 63 (1997) 187 201

was considered by Ramachandran and Rao (1968, 1970) where aj's and bk's are given constants such that the linear statistics are convergent, and the sums may be either finite or infinite. On the other hand, characterizations of univariate distributions by identically distributed linear statistics were established in a series of papers (see, e.g., Davies and Shimizu, 1976; Shimizu, 1968, 1978). In all of these investigations, the

problems were reduced to a functional equation of the form

4)(t) = [ l [O(fi2jt)J'2'[O(-/J:j-I t)] ~':/-', Vt ~ •, (3) /

where O(t )=E[e itX] is the characteristic function of the random variable X to be

characterized, flj's and ",)'s are nonnegative real numbers. Notice that the characteristic function q~(t) in (3) is assumed to be nonvanishing in these investigations. If we let f ( t ) = - I n 0(t), t ~ ~, then Eq. (3) is equivalent to a variant of the integrated Cauchy

functional equation

f ( t ) == ~ f(f12jt)72j + ~ f(--fl2./ l t)v2j T, Vt E ~, (4) J J

and the solution to (4) leads to characterizations of the class of nondegenerate character-

istic functions q~(t), in terms of flj's and Zi's. For discussion and references concerning statistical motivation of the characterization, see Kagan et al. (1973).

Aiming at the multivariate analog of the above, we consider in this paper an inte- grated functional equation in a more comprehensive form on ~n:

f(t)=f0 f ( s t ) d c t ( s ) + j ( ° f ( - s t ) d v ( s ) , V t c ~ ~, (5) ,1] A]

where /~ and v are Borel measures on (0,l]. By deriving the general solutions to the integrated functional equation (5) and some related integral equations, we obtain a general representation for the family of nonvanishing characteristic functions q~(t) such that - ln~b( t ) satisfies Eq. (5). We will show that the family essentially consists of multivariate stable (semi-stable) distributions. Consequently, we have the multivariate generalizations of the above-mentioned characterizations of univariate normal and stable distributions through linear statistics. These results complement previous characteriza- tions for univariate distributions. Our approach is related, in spirit, to that of Davies and Shimizu (1976). We will first establish the form o f - In I(o(t)l = - Re(ln ~b(t)) in Theorem 4.1, and then reconstruct 0( t ) (Theorems 4.3 and 4.8) by using Theorem 3.3 and its corollary. This leads to characterization of multivariate normal and stable dis- tributions. The main idea of the proof is to consider one-dimensional projections and apply the Lau Rao theorem on integrated Cauchy functional equations Lau and Rao, 1982; Ramachandran et al., 1988. It should be pointed out that the L a ~ R a o theorem is a generalization of Deny's (1960) theorem. For recent development on integrated Cauchy functional equation, readers are referred to Davies and Shanbhag (1987), Lau and Zeng (1990), Ramachandran and Lau (1991), and Rao and Shanbhag (1992).

We organize the paper as follows: in Section 2, we discuss some properties of stable and semi-stable distributions, in particular, our remarks concern the stability

.4. K. Gupta el al./ Journal q/Statistical Plannino and In/i'rence 63 (1997) 1,~7 201 I~

of a multivariate distribution versus stability of all one-dimensional projections. In

Section 3, we derive the general solutions to the integrated functional equation (5),

and some related integral equations. We then apply the solutions to obtain in Section 4

a general representation for the family of nonvanishing characteristic functions rib(t),

with - I n qb(t) satisfying Eq. (5). In Section 5, we give multivariate generalizations of

the characterization via linear statistics.

2. Multivariate stable and semi-stable distributions

The stable distributions arise in probability theory as limit distributions for normal-

ized sums of independent and identically distributed random variables 112]. A random

vector X in R" is said to have multivariate stable distribution, if there are sequences

{X,,}, {a,,} and {b,,}, such that {X,,} are independent and identically distributed ~"- t l valued random vectors, a,, c N", b,, >0 , and b,, -I ~ / : I X / - a , , converges to X in dis-

tribution. In terms of characteristic function ~b(t) = E[eit'x], we state the following

necessary and sufficient condition for stability: its proof can be found in any standard

text.

Theorem 2.1. A random vector X is multi~:ariate stable i f and only i/~,/or each c~ > 0

and c: >O, there exist c > 0 and a ~ R", such that the characteristic limctio,l ~h(t) ~!/

X sati.sTfies

~(Cl t)(/)(c2t)

In this case, c is

exponent 0 < c~ <~ 2,

= O(ct)e ia't, ' it ~ []~". ,', 6 )

uniquely determined by c = ( c ~ + c~) 1~, with the characturixtic

independent oj'cl and c2.

If for every pair of cl and e2, we always have a = 0, then Eq. (6) becomes

(/)(Clt)Cfl(c2t) = ~b(ct), g t ~: ~". 7)

Such an X is said to have strictly stable distribution, Clearly, symmetric stable distri- butions are strictly stable. From the derivation of stable distributions, the class perforce

includes normal distributions as distinguished members. In fact, the characteristic expo-

nent ~ 2 implies normality. Hence, stable distributions provide a natural generalization

of normal distributions, and have a significant role in stochastic modeling.

It is known that if X has a multivariate stable distribution, then the one-dimensional projection a 'X in any direction of a,~ JR", a ~ 0, has a univariate stable distribution with the same characteristic exponent :~ of X. One naturally expects the converse holds

as for the normal distributions, and the answer is affirmative if either the character- istic exponent e~> 1, or all the one-dimensional projections a 'X , a 5 0 , are umvari-

ate strictly stable. These are immediate, except for the case c~ = 1, which was settled

in Samorodnitsky and Taqqu (1990). However, a counterexample by Marcus (1983)

190 A.K_ Gupta et al./Journal of Statistical Planning and Injerence 63 (1997) 187-201

shows that the converse is false for 0 < ~ < 1. In light of this fact, the characterization problem we consider here cannot be trivially reduced to the univariate case.

Another closely related class of distributions is the semi-stable distributions. It is sometimes called self-similar distribution. A random vector X is semi-stable if its characteristic function q~(t) satisfies

O(t) = [q~(flt)] ~', Vt ~ ~n, (8)

for some 7>0, and 0 < l f l [ < l . The unique number 2 such that 7[flI;= l plays a role similar to the characteristic exponent ~ for stable distributions. Semi-stable distributions are infinitely divisible, and share a similar L6vy representation as stable distributions. The readers are referred to Jajte (1977), Kruglov (1972) and L6vy (1954) for more details.

3. Solutions to the integrated functional equations

In this section, we will derive the general solutions to the integrated functional equation (5), as well as a related integrated functional equation with signed measure. For our purpose, we only consider continuous solutions. Throughout the rest of the paper, we let S denote the unit sphere in ~", S = {t ~ ~": Iitil = I), where Iltll =(t~ + t~ + .. . + t2) 1/2. For a given Borel measure /t, let supp/~ denote the support of/~.

Theorem 3.1. Let f : ~n___+ • be a nonnegative continuous function, satisfy&g

f ( t ) = /( ° f ( s t ) d l ~ ( s ) + f( ° f ( - s t ) d v ( s ) , V t E ~ n, ,1] ,1]

where I~ and v are Borel measures on (0,1]. Then f is o f the form

t t

where ~ is determined by' f~o, lj s~ d~(s)+f~0,1j s~ dr (s ) = 1, and the functions p, q : S x ~+ -+ ~ are continuous, and satisfy ( l ) p # nonnegative, p(t, .) = p ( - t , .), and p( . ,sr) = p(. ,r), for each s c supp(t~+ v); (2) q(t, .) = - q ( - t , .), and

q( . , sr )= { q(-,r) if s E supp/~,

- q ( . , r ) i f s E supp v.

Proof. Let f : ~n --~ ~ be a nonnegative continuous solution to the integrated functional equation (5). For each to E S, let g(to, r ) = f(r to) , r C R+, then

g(to, r) = /0.1] g(t°'sr)dl~(S)+~o,,] g(-to, sr)dv(s), Vr C ~+. (10)

A.K. Guptel el al. / Journal O! Statistical Plannin{l and h!li'rence 63 (1997) 1,~'7 2()I t91

By letting F(x )=g( to , e ~) and G ( x ) = q ( to, e ~), Eq. (10) is equivalent to the simultaneous integral equations

~0 7yC /0 ̧ x F(x) = F(x + y ) d f i ( v ) + G(x + y )d i~(y),

G(x) G(x + y) dfi(y) + F(x + y ) d f ( y ) ,

( 1 1 )

for all x ¢ A , where dfi(y) d/~(e "), and d f ( y ) = d v ( e - ~ ' ) . Since F and G are both continuous functions, and fi and /~ are Borel measures on A_, Theorem 1 in Ramachandran et al. (1988) yields that

F(x) [p0(e -x ) + q 0 ( e ")]e ~', G ( x ) = [ p 0 ( e x) q0(e ~)]e ~',

where ~ is uniquely determined by , ~ e ~x d(fi + i ) (x) ,11o,~] s~ d ( / , + v)(s) = 1 (it" no such 7 exists, Eq. (1 l) has only zero solution), p0(sr) = p0(r), for each s c supp(/ t+v) , and

f q0(r) if s C supp It, qo(sr)

-qo ( r ) if s ~ suppv.

It follows that

t t t t

where p and q are continuous functions with the desired propemes. ~]

Remark . It should be pointed out that in most of the interesting cases, the solution / in the above theorem takes a much simpler form f ( t ) = p(t/ j j t H )Ut]l ~, with p an even function on S and q ~ 0. To see this, let {suppft) denote the subgroup generated by supp/~ in the multiplicative group (0,,~o). Then we observe the following: (1) If there is s E ( s u p p t t FI (suppv), then q ( . , r ) = q ( . , s r ) q(. ,r), so that q-~().

In fact, this is true whenever (supp p) F/(supp v) ¢ (a. (2) If (supp(/~+v)) is not a subset of {pk,k C ~} for some p > 0 , then it can be shown

that p ( . , s ) = p( . , r ) , for all r , s ¢ R~, so p is a continuous function on S.

Corollary 3.2. Let f be a nonnegative continuous solution o/'the integratedJimetioi,M equation (5). I f f is" an even function, then

f ( t ) - - p ,lltll Iltll ~, 'v ' tc~", (12)

where Jio.ll s~ d(,u + v) (s)= 1, and p( . ,s r )= p(-,r) , fin" each s ~ supp (It + v).

192 A.K. Gupta et al./Journal of Statistical Plannin# and ln[erence 63 (1997) 187 201

Proof. It follows from Theorem 3.1 that f is o f the form

Since f is an even function, we have q(-t/[[t][,.) - q ( t / [ [ t l [ , . ) - - q ( t/Ht[I,. ), so

that q_-0 . []

In deriving the characteristic functions from Eq. (5), we need bounded solutions to

an integrated functional equation with signed measure.

Theorem 3.3. Let # and v be (strict) sub-probability measures on (0,1], such that It + v is a probability measure on (0,1], and let L: ~n ~ R be a bounded continuous

junction, satisfyin9

L(t)= [" L(st)d(It-v)(s), Vt~R", (13) J~0 ,1]

then either one of the J~dlowin9 holds: (1) I f there exists p > 0 such that

supp It C {p2, p4, p6 . . . . }, supp v C {p, p3, p5 . . . . },

then L(pt) = L(t), k/t ~ ~'L

(2) L =_ O, otherwise.

Proof. Eq. (13) implies that for each to E S ,

L(rto)= / ' L (sr to )d( I t - v)(s), VrE ~+, ,.t (0,1]

and hence, by Theorem 3 in Ramachandran et al. (1988),

= f L(rto) if s C supp It, L(srto )

( L(rto) if s E supp v.

It follows that L ~ 0 unless supp (# + v) is a subset o f a geometric sequence {pk} with suppItC_ {p2,p4, p6 . . . . ), and suppvC {p, p3,p 5 . . . . }. []

Following the same lines as in Ramachandran et al. (1988, p. 195), we have:

Corollary 3.4. The conclusion o f Theorem 3.3 holds if for each to c S, and 0 < e < 1, the function L(ra to) - L(rto) is bounded for r>~O, while the Eq. (13) holds (L itself need not be bounded).

A.K. Gupta et al./Journal q[" Statistical Planning! and In[~'rence 63 (1997) L~7 2(11 1~3

4. Characterist ic functions satisfying the integral equation

Consider nonvanishing characteristic function $(t), with In qb(t) satisfying the inte- grated functional equation (5),

, l ] , Ltj

Theorem 4.1. Let ~b(t)= [~h(t)l 2. Then ~)(t) is a symmetric semi-stable characteristic junction qf the ,[arm

~,(t) exp [ - 2 p ( ~ , ],t,,),,t,]~] , (15)

where .]i0,11 s~d(t~ + v)(s) = 1, and p : S x ~+ ~ U~+ is a bounded continuous/Uncti,m lhat sati,s;fies p(t , . ) p ( - t , .), and p ( . , s r )= p(.,r), ,h~r each s c s u p p ( t / + v).

In particular, (/ supp(ld + v) is not contained in a set {pk ,k - -1 ,2 . . . . } , /m" s,mle p ~ (0, 1 ), then ~9(t) is stable with p(t, .) a constant junction jor each t ~ S.

Proof. Since ~9(t) = I~h(t)l 2 is a symmetric characteristic function, In ~9(t) - 2 In ](/~(t)] is an even function, and satisfies

l n , ( t ) f l n * ( s t ) d ~ ( s ) + f l n , ( - s t ) d v ( s ) , V t ~ " . (16) .ll~ h ~ ) , l ] . ~.11

It follows from Corollary 3.2 that

In O(t) = 2p(l~t~,l l t l[)Ht, lL

where .[10,tl S~ d(tt + v)(s) = 1, and p : S × N+ ~ [~ ~ is a continuous function satisfying p ( t , . ) = p ( t , . ) , a n d p ( . , s r )=p ( . , r ) , for e a c h s ~ s u p p ( / ~ + v ) . Hence, for each t ~ S , p( t , r ) is a bounded function of r on [ 0 , ~ ) . The compactness of S and continuity of p then implies that p is bounded.

Notice that, for fi ~ supp (tL + v),

) In ~(fit) = 2p \ II/~tll' Ilfitll Ilfitll ~ /3~ ln,p<t).

There exists 7 > 0 such that 7fi ~= 1, and hence , , , In~(f i t )=Tfi~ln~(t)= In ~//(t), or ~b(t) [~p(flt)]:', so ~b is semi-stable.

In case s u p p ( t z + v ) is not contained in a set of geometric sequence {p~,k 1,2 . . . . } for some p > 0 , p(t, ,) is a constant function for each t ~ I~. This gives

In 0 ( , ) = - p Iltll =,

and ~( t ) is stable with characteristic exponent 7.

194 A.K. Gupta et al./Journal of Statistical Plannin9 and Inference 63 (1997) 187-201

It follows from Theorem 4.1 that the characteristic function ~b(t) is o f the form

I ( + ) ] q~(t)= exp - p ,lltll Iltll ~ + i A ( t ) , (17)

where A(t) is an odd function to be determined. To recover ~b(t) from @( t )= ]qS(t)] z,

we will consider two cases where ~ ~>2, and 0 < ~ <2 .

Theorem 4.2. Let ~b(t) be a characteristic function that satisfies the integrated func- tional equation (14). (1) I f ~>2, then ~(t) is' degenerate. (2) IJ'~ = 2, then (~(t) is multivariate normal (possibly degenerate).

Proof . Let X be a random vector in ~n with characteristic function ~9(t) as in (15).

For to C S, the univariate characteristic function ~t~x(r) of t~X is

~/t;x(r)---- tp(rto) = exp[-2p( t0 , Irl)lr[~], Vr E ~, (18)

which is univariate semi-stable. It is well known that if ~ is the characteristic exponent

of nondegenerate univariate stable (semi-stable) distribution, then 0 < ~ ~<2, with nor- mality when c~ = 2 . Thus, (1) follows, since ~ ( t ) is degenerate in case 7 > 2 . I f ~ = 2 , then t6 X is normal for every to E S, hence ~ ( t ) is multivariate normal. The Cramer 's Theorem for multivariate normal distributions yields the normality of ~b(t).

Let 0 < ~ < 2. We will prove our main theorems in the following by reconstructing the characteristic function q~(t) from @ ( t ) = I~b(t)l 2.

Theorem 4.3. Let ~(t) be a nonvanishin9 characteristic function that satisfies the integrated functional equation (14), with 0 < : ¢ < 2 . I f supp(# + v) is not a subset o f {pk, k = 0 , 1 , 2 .... } for any p > 0 , then q3(t) is the characteristic function of a multivariate stable distribution.

We first prove a lemma.

L e m m a 4.4. The characteristic function ~(t) in (15) satisfies

I1 - ~ b ( t ) l = O ( l l t U ~ ) , a s t - ~ o .

Proof . By Theorem 4.1, ~9(t) is o f the form

~(t)= exp[-2P ( ~tu, llt") "t'[~ ] ,

where p : S x ~ + - ~ [~+ is a bounded continuous function. Let M > 0 be such that O<~p<~M. As t---~0,

0 ~< l1 - @(t)l 1 - e x p [ - 2 p ( II~H, Iltll )lltlI ~] 1 - e -2MIItlF - - ~< < ~ .

Iltl[ ~ I[tll ~ Iltll =

We have I 1 - @(t)I =o(l[tl]:~), as t---~o. []

A.K. Gupta et aL /Journal ~/' Stati~+tical Phmnin{t and b~l~'rence 63 (1997) 1,%'7 2(11 195

In order to obtain the general lbrm of ~b(t), we require the following theorems

concerning the relation between the analytic properties of a characteristic function and

tail behavior of the corresponding distribution. We state the multivariate version of the

theorems due to Bai and Miao (1984).

Theorem 4.5. For r ~ (0, 1), the Jollowinq are equit:alent: (1) P[ILXLI > u] = O(U--(m+r)), a S U---+ :SX),/()r some inte~,ter rn ~>0;

(2) There exists a polynomial Qm(t) such that

~(t) Qm(t)+O(lltll ''+'+) as t +0,

where X is a random eector in ~" with characteristic/unction ~(t), and

- ~ i~--- [ (t,x) k dF(x) , Qm(t) k=o k! j~,,

with <t,x) the inner product of t and x, and F the distribution function +~/ X.

Theorem 4.6. For a fixed integer m >7 O,

P[!lXl{>, ,] = o ( . -~m+~) as ,, ~ ~ ,

( /and onO~ ([+ there exists a polynomial R2m(t) such that

Re ~ ( t ) = R2m(t) + O(Htll (2m~ 1)) as t ~ o,

where R2m(t) Re(Q2m(t)), with Q2m as in Theorem 4.5, and Re ~(t) is the real part

o f ~(t).

It [bllows from the above theorems that we have

Lemma 4.7. Let 0 < ~ < 2, and let X be the random vector with characteriftic fimction 4~(t), satisf/vinq the integrated fimctional equation (14). Then

P[I IXH>u]=O(u -~) as u - -~oc . (19)

Proof. Let X and Y be independent and identically distributed random vectors in !~", with characteristic function qS(t). Then 0 ( t ) = 14~(t)j 2 is the characteristic function of

X - Y. Lemrna 4.4 together with Theorem 4.5 (for the case ~ ¢ 1) and Theorem 4.6

(for the case :~ = 1) imply that

P[IIX - YII >u] = O(u ~) a s . -~ ~

Let 7 > 0 be the median of the univariate random variable II YI[, then

P[I IlXlL - I[ v i i i > u ] >t ½P[IlXll > u + ~,q

Since e [ l I X - YII >ul~>P[I IIXll IIYII I>u], for large u,

2P[IIX Y l l > u - 7 ] > ~ P [ l t X l l > u ] ,

196 A.I( Gupta et al./Journal of Statistical Planniny and InJerence 63 (1997) 187-201

and hence, as u ---+ oc,

P[[IXll>u] ~2P[II X - YII > u - T J ( u - 7 ) -~ <~ K, u - ~ ( u - ~ , ) - ~ u

for some 0 < K < o c , i.e., P[llXll > u ] - O ( u ~).

P roof of Theorem 4.3. We divide the proof into two parts, where ~ ¢; 1, and :~ = I.

Part 1: ~ ¢; 1. From Lemma 4.7 and Theorem 4.5 there exist mj,m2 . . . . . mn such

that

" ~ ( t ) l + i ~ mktk = O(I]tH ~) as t --+ 0, (20) k=l

where r n k - 0 , k - 1 , 2 , . . . , n , if 0 < : ~ < l , and the integrated functional equation (14)

implies that mk 7 k 0 only if Ji0,1] s d#(s) - f(0,,] s d v ( s ) = 1. In light o f (20), we can

write qS(t) as

[ ] q~(t) = exp - p , li t Iltll ~ + id( t ) , (21)

where A ( t ) = ~'~=, mkt~ + IItll~l(t), l ( t ) is a bounded continuous function, and - I ( t ) = I ( - t ) . It follows from Eq. (14) that

l ( t ) = [ I (s t )s ~ d ( # - v)(s), Vt C •n. (22) .J(o ,I]

There are three cases to be considered:

(1) I f v-= 0, then Eq. (22) reduces to

I ( t ) = f I(s t)s ~ d/~(s), Vt E E ~. (23) J(0 ,1]

It follows from the Lau Rao Theorem [10] that I ( t ) = I ( s t ) , for each s C_supp#,

and t E N". (2) I f # and v are both nonzero, and there exists p > 0 such that

supp#C{p2, p4, p 6 . . . . }, suppvC_{p, p3,p 5 . . . . },

then, Theorem 3.3 implies that I (pt ) = I(t) , Vt E R ~. (3) If /~ and v are both nonzero, and no such p > 0 exists as in case (2), then

Theorem 3.3 again yields that I -= 0. Now, since supp(/~ + v ) = (supp#)tO (supp v), and supp(# + v) is not a subset of

{ p k , k = 0 , 1 , 2 . . . . } for any p > 0 , the function p( t , . ) in (21) is a constant function

for each t E S , and for the function I(t), only case (1) and case (3) are possible. In case (1), the function I ( t ) satisfies that l ( t ) = I(st) , for s E (0, oc), so it is simply a

continuous function on S, i.e. I ( t )=I( t / l l t l l ) with I ( - t ) = - I ( t ) . In case (3), I _= 0. Thus, the characteristic function qS(t) is o f the form

qS(t) = exp - p Iltll ~ + i mktk + I Iltll ~ , (24)

A.K. Gupta et al./ Journal qf Statistical Pl~mnin# and h~A'rence 63 ,'1997 ) 1+~¢7 20] 197

where p , l :S---+N are continuous functions. For any a~ > 0 , a2 > 0 , choose a such that

a~+a~ a ~, and c = a l + a 2 - 1 , then

~/~(alt)O(azt ) ~ ( a t ) e x p l i c k ~ mkt~l , (25)

so the stability of qS(t) follows.

Part 2: ~. 1.

By direct computation similar to Davies and Shimizu (1976) we have, for any given

+; with 0 < ~ : < 1 , and t 0 ~ S ,

I - qS(rt0)_ 1 qS(~:rt0) = O(~:-1), (26) I" EF

as r -+ 0. This implies that for each ~:,

1 - q ~ ( t ) 1 0 ( g , t )

is a bounded function of t E ~", and hence c/~(t) can be written as (21), with ma =:0,

and the function I ( t ) such that for each to E S, 0 <~; < 1, the function I(m:to) l(rto) is a bounded function of r > 0 . Moreover, the function l ( t ) satisfies

l ( , ) /" l(st)s~ d(l, - v)(s), V i c e " . ,]I~3.1]

The Lau-Rao Theorem (Kagan et al., 1973) and Corollary 3.4 lead to three cases

similar to the above:

(1) If v ~ 0, then l ( t ) = l ( s t ) , for each s 6 suppp, and t ¢ R".

(2) If I~ and v are both nonzero, and

supp It C_{p 2, [)4 1)6 . . . . }, supp V C{p, 1> 3, p5 . . . . },

for some p > 0 , then l ( p t ) = l ( t ) , t C ~". (3) 1 ~ 0, otherwise.

The same argument as in the case ~ ¢ 1 yields that 0 ( t ) is of the form

[(+) +(t) exp - p tltll ;l Iltll {27)

and the stability of qS(t) follows immediately.

In the proof of Theorem 4.3, the forms of the characteristic function c/~(t) are given

in three cases, according to the structure of supp(IL + v). We point out that if supp

( t t + v ) is a subset of a geometric sequence {pa, k = 0, 1,2 . . . . }, then 0 ( t ) is semi-stable,

except tbr a few cases, as shown in the following theorem.

Theorem 4.8. Let O(t) be the e/Taracteristic /}ruction, sati~J.i,im I the inte~lrated /imc- tional Ee I. (14), and suppose there is no p > 0 such that

supppC{p2,p4 ,p 6 . . . . }, suppvC_{p, ps,p 5 . . . . },

198 A.K. Gupta et al./ Journal oj Statistical Planning and Injerence 63 (1997) 187 201

(1) I f 0 < ~ l , then O( t ) is semi-stable. (2) I f 1 < c t < 2 , and m ~ = 0 , k = 1,2 . . . . . n, where mk's are given in (20), then O(t)

is semi-stable.

Proof. It follows from the assumption that only cases (1) and (3) in both parts of the

proof of Theorem 4.3 are possible. I f 0 < c~ ~< 1, then mk = 0 and ~b(t) can be written

as

~b(t) = exp - p ]lt]] ~ + i I ]]t[I ~ , t c R", (28)

where p(. , ]]fit]]) = p(., lit]I), I(flt) = I(t), for each/3 C supp(u + v), and t E ~". Hence,

~b(flt) = exp{[:/~ [ - p (l~ll)Ittll~ + il (ll~ll) ,[tl,~] }. With 7 > 0 such that ~,/~ = l, we have O(flt)~':O(t), and the semi-stability of O(t) follows.

I f 1 < ~ < 2 , and mk=O, k = 1,2 . . . . ,n, then ~b(t) is of the form in (28), and the semi-stability can be shown similarly.

5. Characterization through linear statistics

As a consequence of our main theorems, generalization of characterization of univari- ate distributions through linear statistics to multivariate distributions can be obtained. We conclude this final section with two such generalizations.

5.1. Identically distributed linear statistics

Let XI, X2 . . . . be independent and identically distributed nondegenerate random vec- tors in R ' , and {aj} be a sequence of real numbers such that ~ j ajXj converges almost surely.

I f for some r, rX1 and ~ j ajXj are identically distributed, without loss of generality, we may assume r = 1 and ]aj[ ~< 1, then the characteristic function qS(t) o f Xj satisfies, by rearranging aj's if necessary,

cb(t) I](a(a2jt)O(a2j_lt), Vt 6 JR', (29) J

where O<~a2j<l , and - 1 <a2j-1 ~<0. It is clear that ~b(t) is nonvanishing, and hence Eq. (29) is equivalent to

In qS(t) = ~ In O(a2jt) + ~ In (p(a2j-it). J J

(30)

By our main theorems in Section 4, the distribution of X1 is characterized as: ( l ) XI has a multivariate normal distribution if and only if ~-]j a~ = 1 (Theorem 4.2).

A.K. Gupta et aL /Journal of Statistical Plannm 4 am/ h~/brem'e 63 (1997) 187 201 199

(2) Let 0 < ~ . < 2 be uniquely determined by ~ ' i laj]~ = 1.

(a) If {la/I} is not a subset of any geometric sequence, then X1 has a multivariate stable distribution (Theorem 4.3).

(b) If {la/I} is contained in a geometric sequence, but there is no y > 0 such that a2j c {p2, [)4, p6 . . . . }, ]a2/ _11~ {O, [,)3 f,~5 . . . . }, then X~ is multivariate semi-

stable, except possibly for 1 <,z~<2 and ~ ' j a~ 1 (Theorem 4.8). We remark that Theorems 4.1 and 4.2 in Gupta et al. (1992) can also be derived

fi-om the above results.

5.2. Zero re~jre,s'sion of one linear statistic otl another

Let Xl, X2 .... be a sequence of independent and identically distributed random vec- tors in JR", with E[XI] = 0 . Let {a/} and {b/} be sequences of real numbers such that ~ / a i X / and y~ ' /b iX I converge with probability one, and

E [~a ,X i ~b/X/1 = 0 , a.e. (2,1) L / / j

The distribution of X 1 depends on the nature of the coefficients a / ' s and bi's and their relation, Without loss of generality, we may assume 0 < Ib/I < 1, and consider

Furthermore, we suppose a//bi >0. (This is automatically satisfied if X~ is symmetric. ) Then for t ~ JR',

With (/)(t denoting the characteristic function of XI, we have

E r v it'Xi it~X, /~il f? ] ~ 05(bkt) = ~ ~ +(bk a/E[X/e ' ](/)(t) t), / I,/!

and hence, for 1 ~< m ~< n,

,'<1, ~(b j t ) U, (33)

4,(t) / - ( ~ ( b i t ) '

where U is a neighborhood of 0. Integrating (33) leads to

qS(t) = I lq~(b/ t ) < / ' ' , t c U, (34) /

where 0 < Ib, l < 1, and aj/hj>O. This reduces the regression equation (31) to the inte- grated functional equation (14). Note that E[XI] 0, we have:

200 A~K. Gupta et al . /Journal o f Statistical Planning and In]brence 63 (1997) 18~201

(1) XI is normally distributed if and only if }-~j a/bj = 1 (Theorem 4.2).

(2) I f {Ibjl} is not a subset o f { p k , k : 0 , 1,2 . . . . } for some p > 0 , then Xl has multi- variate stable distribution, with characteristic exponent ~ uniquely determined by ~/la/jrbjl ~-I = 1 (Theorem 4.3).

(3) I f {Ib/I} is a subset o f a geometric sequence, but there is no p > 0 such that b.i C {p2,p4, p6 . . . . }, for by)O, and bj ~ {p, p3,pS,...}, for tS/~<0 , then Xi is mul-

tivariate semi-stable, except possibly for 1 < :~ < 2 and ~ / a / = 1 (Theorem 4.8).

Acknowledgements

The authors are indebted to Professors C.R. Rao and Ka-Sing Lau for bringing their attention to the problem, and for the invaluable discussions. The authors also thank referees for their helpful comments and suggestions.

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