ON A CONDITIONAL CAUCHY FUNCTIONAL EQUATION OF SEVERAL VARIABLES AND A CHARACTERIZATION OF MULTIVARIATE STABLE DISTRIBUTIONS

The general solution of a conditional Cauchy functional equation of several variables is obtained and its applications to the characterizations of multivariate stable distributions are studied.


INTRODUCTION.
The purpose of this note is to solve the conditional Cauchy functional equation of several variables f(ax) aaf(x), 'x e R", (1.1) for a continuous function f on Rn, where c > 0 and a 2,3.
We apply solutions of the equation (1.1) to problems of characterization of multivariate stable distributions, and generalize a result of Eaton [1 on characterization of univariate stable distributions based on a functional equation. 2

. MULTIVARIATE STABLE DISTRIBUTIONS
A random variable with distribution function F is said to have univariate stable distribution if for every bl,b2 > 0, cl,c2 .R, there correspond a b > 0 and c R such that where * denotes the convolution operator of distribution functions.Lvy [4] showed that a univariate stable distribution has characteristic function t) of the form where the constants y,/1, ot satisfy the conditions '> 0, 1/11 a 1, 0 < a 't e Rn, where 0 < a < 2, /t Rn, gl and g2 are continuous functions on R n given by integral representations (Lvy [5]).We remark that the explicit algebraic representations of g! and g2 for multivariate stable distributions obtained by Press [7] is not for all multivariate distributions.It is complemented by examples in Paulauskas [6], indicating that the closed expressions of gl and g2, analogous to univariate case, are still unknown.

THE SOLUTIONS OF A CONDITIONAL CAUCHY FUNC'I'IONAL EQUATION
OF SEVERAL VARIABLES.
In this section, we will derive the general solution f of the Cauchy-type functional equation (1.1), where at > 0 is given, and a 2,3.
Let f: R n R be continuous solution of the equation (1.1). ,On each Ga, we have f(ars) aaf(rs), Vr > 0, where tx > 0, and a 2,3.By letting fs(r) f(rs), from Eaton (1966) and Gupta et al. (1988) on solutions of a Cauchy equation of one variable, we get f(r) f(1)r u, ,'r > 0.
It follows that for x Rn, x e 0, If we let g(s) fs(1) where s is an element of S, then g is a continuous function on

S and
We obtained the general solution of the equation (1.1).THEOREM 3.1.The general continuous solution f of the conditional Cauchy functional equation (1.1) is of the form where g is an arbitrary continuous function on S.
An equivalent form of Theorem 3.1 is THEOREM 3.1'.The general continuous solution f of the conditional Cauchy functional equation f (a-x) af (x), Vx Rn, where o > 0, and a 2,3, is given by where g is a continuous function on S. PROOF.We need only show the sufficiency.Let be the characteristic function of XI.We first show that (t) 0, Vt Rn.Suppose there is to R n such that t0) 0. Then the characteristic function of the univariate random variable tXl is __txl(U) uto), with __txl(1) t0) 0. In the proof of Theorem 3.1 of Gupta et al.
(1988), it is shown that if __#t6Xl (1) 0 then tbxl(u) 0 for all < u < and then 0) 0 which is a contradiction.Hence can be written as hl(t)+ih2(t) (t) e or equivalently as, .n(t)= hi(t) + ih2(t), where hi and h2 are real-valued continuous functions on Rn, the former is even and the latter is odd.
The hypotheses imply that for some 0, 0 < a < 2, hl(al/at) ahl(t) and h2(a ah2(t), 't Rn, a 2,3, and hence by Theorem 3.1', hi and h2 are of the form and Therefore by (2.2) X1 has a multivariate stable distribution with location parameter vector / 0.
In the case o 1 to show that all the univariate marginals of X1 are Cauchy distributed, we are going to show that the first component of X1 is Cauchy distributed, the Cauchy distributed of the other components of X1 are obtained by a similar way.
Denoted by el a vector of R n having the first component equal to 1, and the other components equal to zero.The characteristic function l of the first component is obtained by giving wel in the characteristic function of X1, for every w R, that is, n l(W)= n we,> I1, By he fac ha ] s an even function, 2 Js an odd function in he n (-D 1(0, 2(-0 -2(0, respectively, for every n @(w) w[gl(el) + ig2(el)] wgl(el) + iwg2(el), and if w < 0, n (w) -wl(-el) + ig2(-el)] -wl(el)-ig2(el)] -wgl(el) + wg2(el).Therefore for any w R, n (w) w gl(el) + iwg2(el), and 1 is the characteristic function of a Cauchy disibution.As mentioned ave, Press [7] showed that for any mulvariate stable disibufion, the characteristic function defined by (2.2) can given by a simple explicit algebraic form, but Paulauska [6] inted out that the form given by Press is only ue for a class of multivariate stable disibutions, and that the cl forms of gl and g2 are still unknown, except for the case a 2, g2(t) 0, and g(t) t't for some sifive definite n x n matrix .THEOM 4.2.t X,X2 X9 indendent and identically disibut random vectors in Rn.
(ii) X has a multivariate stable distribution with Cauchy marginals if and only if l d X + X 3Xl and Xl + X2 + X3 are identically disibuted, resctively.
PROF. e prf follows along the exact same lines as that of eorem 4.1 by using rem 3.1 with a 1 or 2, instead of Threm 3.1'.
ACKNOWLEDGEMENT.The authors thank the referees for the useful comments and suggestions.
4. A CHARACTERIZATION OF MULTIVARIATE STABLE DISTRIBUTIONS Applying Theorem 3.1', we get the following characterization of multivariate stable distributions.THEOREM 4.1.Let XI,X2,X 3 be independent and identically distributed, nondegenerate random vectors in Rn.Then XI has a multivariate stable distribution with / 0 in (2.2) if and only if there is an , 0 < o < 2, such that 21/aXl and XI + X2, 31/aXl and XI + X2 + X3 are identically distributed, respectively.In case a 1, all the univariate marginals of XI are Cauchy distributed.