POSITIVE DEFINITE FUNCTIONS AND MULTIDIMENSIONAL VERSIONS OF RANDOM VARIABLES

POSITIVE DEFINITE FUNCTIONS AND MULTIDIMENSIONAL VERSIONS OF RANDOM VARIABLES ALEXANDER KOLDOBSKY Abstract. We prove that if a function of the form f( K is positive definite on, where K is an origin symmetric star body in and f : R R is an even continuous function, then the space (, K embeds in L. We apply this result to the problem of characterization of n-dimensional versions, i.e. random vectors in such that all linear combinations of their coordinates have equal distribution, up to a scalar constant. Combined with known facts about embedding in L, this result also leads to several generalizations of the solution of Schoenberg s problem on positive definite functions. 1. Introduction Following Eaton [E], we say that a random vector X = (X 1,..., X n is an n-dimensional version of a random variable Y if there exists a function γ : R, called the standard of X, such that for every a the random variables n a i X i and γ(ay (1 i=1 are identically distributed. We assume that P {Y = } < 1. A problem posed by Eaton is to characterize all n-dimensional versions, and, in particular, characterize all functions γ that can appear as the standard of an n-dimensional version. It is easily seen [M3], [Ku] that every standard γ is an even homogeneous of degree 1 non-negative (and equal to zero only at zero continuous function on. This means that γ = K is the Minkowski functional of some origin symmetric star body K in. Recall that a closed bounded set K in is called a star body if every straight line passing through the origin crosses the boundary of K at exactly two points, the origin is an interior point of K and the Minkowski functional 1

2 ALEXANDER KOLDOBSKY of K defined by x K = min{s : x sk} is a continuous function on. Note that the class of star bodies includes convex bodies containing the origin. Eaton [E] proved that a random vector is an n-dimensional version with the standard K if and only if its characteristic functional has the form f( K, where K is an origin symmetric star body in and f is an even continuous non-constant function on R (see [K3, Lemma 6.1] for a short proof. By Bochner s theorem, this means that the function f( K is positive definite. Recall that a complex valued function f defined on is called positive definite on if, for every finite sequence {x i } m i=1 in Rn and every choice of complex numbers {c i } m i=1, we have m m c i c j f(x i x j. i=1 j=1 Thus, Eaton s problem is equivalent to characterizing the classes Φ(K consisting of even continuous functions f : R R for which f( K is a positive definite function on. In particular, K appears as the standard of an n-dimensional version if and only if the class Φ(K is non-trivial, i.e. contains at least one non-constant function. In some places throughout the paper we write Φ(E K instead of Φ(K, where E K = (, K is the space whose unit ball is K. The problem of characterization of positive definite norm dependent functions has a long history and goes back to the work of Lèvy and Schoenberg in the 193s. Lèvy [Le] proved that, for any finite dimensional subspace (, of L q with < q 2, the function g(x = exp( x q is positive definite on, and any random vector X = (X 1,..., X n in, whose characteristic functional is g, satisfies the property (1. This result gave a start to the theory of stable processes that has numerous applications to different areas of mathematics. The concept of an n-dimensional version is a generalization of stable random vectors. In 1938, Schoenberg [S1,S2] found a connection between positive definite functions and embeddings of metric spaces in Hilbert spaces. In particular, Schoenberg [S1] posed the problem of finding the exponents < p 2 for which the function exp( x p q is positive definite on, where x q = ( x 1 q +... + x n q 1/q is the norm the space l n q with 2 < q. This problem had been open for more than fifty years. For q =, the problem was solved in 1989 by Misiewicz [M2], and for 2 < q <, the answer was given in [K1] in 1991 (note that, for 1 p 2, Schoenberg s question was answered

POSITIVE DEFINITE FUNCTIONS 3 earlier by Dor [D], and the case n = 2, < p 1 was established in [F], [H], [L]. The answers turned out to be the same in both cases: the function exp( x p q is not positive definite if n 3, and for n = 2 the function is positive definite if and only if < p 1. Different and independent proofs of Schoenberg s problems were given by Lisitsky [Li1] and Zastavnyi [Z1, Z2] shortly after the paper [K1] appeared. For generalizations of the solution of Schoenberg s problem, see [KL]. The solution of Schoenberg s problem can be interpreted in terms of isometric embeddings of normed spaces. In fact, the result of Bretagnolle, Dacunha-Castelle and Krivine [BDK] shows that a normed space embeds isometrically in L p, < p 2 if and only if the function exp( p is positive definite. Hence, the answer to Schoenberg s problem means that that the spaces l n q, q > 2, n 3 do not embed isometrically in L p with < p 2. The classes Φ(K have been studied by a number of authors. Schoenberg [S2] proved that f Φ(B2 n if and only if f(t = Ω n (tr dλ(r where B2 n is the unit Euclidean ball in Rn, Ω n ( 2 is the Fourier transform of the uniform probability measure on the sphere S n 1, and λ is a finite measure on [,. In the same paper, Schoenberg proved an infinite dimensional version of this result: f Φ(l 2 if and only if f(t = exp( t 2 r 2 dλ(r. Bretagnolle, Dacunha-Castelle and Krivine [BDK] proved a similar result for the classes Φ(l q for all q (, 2 (one just has to replace 2 by q in the formula, and showed that for q > 2 the classes Φ(l q (corresponding to infinite dimensional l q -spaces are trivial, i.e. contain constant functions only. Cambanis, Keener and Simons [CKS] obtained a similar representation for the classes Φ(l n 1. Richards [R] and Gneiting [G] partially characterized the classes Φ(l n q for < q < 2. Aharoni, Maurey and Mityagin [AMM] proved that if E is an infinite dimensional Banach space with a symmetric basis {e n } n=1 such that e 1 + + e n lim =, n n 1/2 then the class Φ(E is trivial. Misiewicz [M2] proved that for n 3 the classes Φ(l n are trivial, and Lisitsky [Li1] and Zastavnyi [Z1], [Z2] showed the same for the classes Φ(l n q, q > 2, n 3. One can find more related results and references in [M3], [K3].

4 ALEXANDER KOLDOBSKY In all the results mentioned above the classes Φ(K appear to be nontrivial only if K is the unit ball of a subspace of L q with < q 2. An old conjecture, explicitly formulated for the first time by Misiewicz [M1], is that the class Φ(K can be non-trivial only in this case. In support of this conjecture, there are several results available where this is proved under extra conditions on the function f; see [M1], [Ku], [K4]. These conditions assume that f or its Fourier transform have finite moments of certain orders at zero or at infinity. A slightly weaker conjecture was formulated by Lisitsky [Li2]: if a non-constant function f( K is positive definite on, then the space (, K embeds in L. The concept of embedding in L was introduced and studied in [KKYY], the original conjecture of Lisitsky was in terms of the representation (2. Definition 1. We say that a space (, K embeds in L if there exist a finite Borel measure µ on the sphere S n 1 and a constant C R so that, for every x, ln x K = ln (x, ξ dµ(ξ + C. (2 S n 1 The conjecture of Lisitsky was confirmed in [Li2] and in [K4] under extra assumptions of moment type on the function f or its Fourier transform. For example, the result from [K4] is that if f( K is a positive definite function and lim t t f(t = for some >, then the space (, K embeds in L. In this article we prove the conjecture of Lisitsky in its full strength: Theorem 1. Let K be an origin symmetric star body in, and suppose that there exists an even non-constant continuous function f : R R such that f( K is a positive definite function on. Then the space (, K embeds in L. Corollary 1. If a function γ is the standard of an n-dimensional version of a random variable, then there exists an origin symmetric star body K in such that γ = K and the space (, K embeds in L. In the last section of the paper we use known results about embedding in L to provide rather general classes of normed spaces for which the classes Φ are trivial and whose norms cannot serve as the standard of an n-dimensional version.

POSITIVE DEFINITE FUNCTIONS 5 2. Proof of Theorem 1 As usual, we denote by S( the space of infinitely differentiable rapidly decreasing functions on (Schwartz test functions, and by S ( the space of distributions over S(. If φ S( and f S ( is a locally integrable function with power growth at infinity, then f, φ = f(xφ(x dx. We say that a distribution is positive (negative outside of the origin in if it assumes non-negative (non-positive values on non-negative Schwartz s test functions with compact support outside of the origin. The Fourier transform of a distribution f is defined by ˆf, φ = f, ˆφ for every test function φ. We use the following Fourier analytic characterization of embedding in L proved in [KKYY, Th.3.1]: Proposition 1. Let K be an origin symmetric star body in. The space (, K embeds in L if and only if the Fourier transform of ln x K is a negative distribution outside of the origin in. Now we are ready to start the proof of Theorem 1. Proof of Theorem 1. We write instead of K. By Bochner s theorem, the function f( is the Fourier transform of a finite measure µ on. We can assume that f( = 1, and, correspondingly, µ is a probability measure. The function f is positive definite on R, as the restriction of a positive definite function, therefore, f(t f( = 1 for every t R (see [VTC, p.188]. Let φ be an even non-negative test function supported outside of the origin in. For every fixed t >, the function f(t is positive definite on, so f(t x ˆφ(x dx = (f(t, φ(x. (3 R For any (, 1, the integral ( 1 g( = t 1+ f(t x dt + 1 t 1 f(t x dt ˆφ(xdx converges absolutely, because f is bounded by 1 and the function in parentheses is bounded by 2/. By the Fubini theorem, 1 g( = t (R 1+ f(t x ˆφ(xdx dt n (4

6 ALEXANDER KOLDOBSKY + 1 t (R 1 f(t x ˆφ(xdx dt, n so by (3 the function g is non-negative: g( for every (, 1. (5 Now we study the behavior of the function g, as. We have ( x g( = x t 1+ f(t dt + x t 1 f(t dt ˆφ(xdx + ( x 1 x = t 1+ f(t dt ˆφ(xdx (6 R n + x 1 R n ( x t 1+ f(t dt + x ( t 1 f(t dt ˆφ(xdx (7 x x t 1 f(t dt ˆφ(xdx. (8 We write g( = u( + v( + w(, where u, v, w are the functions defined by (6, (7 and (8, respectively. We start with the function w. Lemma 1. lim w( =. Proof : We can assume that < 1/2. Fix a >. Since φ is supported outside of the origin, we have ˆφ(xdx = and R ( n a t 1+ f(t dt + t 1 f(t dt ˆφ(xdx =, a because the expression in parentheses is a constant. Subtracting this from (8 we get ( x ( w( = t 1+ t 1 f(t dt ˆφ(xdx. Now for some θ( [, 2], a t 1 t 2 1 = 2 t 1 t θ( lnt 2(1 + a 3/2 + x 3/2 ( lna + ln x,

POSITIVE DEFINITE FUNCTIONS 7 so w( 2 x a (1+a 3/2 + x 3/2 ( lna + ln x ˆφ(x dx. The latter integral converges because ˆφ is a test function and decreases at infinity faster than any power of the Euclidean norm, and also the function under the integral is locally integrable if the dimension n 2 (see [K3, Lemma 2.1] for a simple argument showing that x p is locally integrable on for any < p < n. We need the following elementary and well known fact. We include a proof for the sake of completeness. Lemma 2. Let h be a bounded integrable continuous at function on [, A], A >. Then A lim t 1+ h(tdt = h(. Proof : We can assume that < 1. We have = A t 1+ (h(t h(dt + h( The first summand is less or equal to t 1+ h(tdt t 1+ dt + A max h(t h(, as, t [,] t 1+ h(tdt. because h is continuous at. The second summand is equal to h( h(, as. The third summand is less or equal to A max h(t, t [,A] as. Lemma 3. lim u( = f( ln x ˆφ(xdx. Proof : Using the estimates x 1 = 1 x θ ln x dθ ln x (1 + x 1 and x t 1+ f(tdt x x + 1,

8 ALEXANDER KOLDOBSKY we see that the functions under the integral over in u( are dominated by an integrable function ln x (1 + x 1 ( x + 1 ˆφ(x of the variable x on. Clearly, for x, x 1 lim = ln x. Also, by Lemma 2, for every x, x x lim t 1+ f(tdt = f( = 1, so the functions under the integral in u( converge pointwise almost everywhere to ln x ˆφ(x. The result follows from the dominated convergence theorem. Lemma 4. There exists a sequence k such that lim v( k = c ln x ˆφ(x dx k with c < 1. Proof : By a dominated convergence argument, similar to the one used in the previous lemma, it is enough to prove that there exist a sequence k and a number c < 1 such that for every x, x lim k k x t 1 k f(tdt = c. For every x we have x t 1 f(t dt x, as, so it is enough to find a sequence k and a number c < 1 such that where lim ψ( k = c < 1, k ψ( = t 1 f(tdt. Since the function ψ is bounded by 1 on (, 1, it suffices to prove that ψ( cannot converge to 1, as.

POSITIVE DEFINITE FUNCTIONS 9 Suppose that, to the contrary, lim ψ( = 1. We use the following result from [VTC, p. 25]: if µ is a probability measure on and γ is the standard Gaussian measure on, then for every t > µ{x : x 2 > 1/t} 3 (1 ˆµ(tydγ(y, (9 where 2 is the Euclidean norm on. Let µ be the measure satisfying ˆµ = f(. For every (, 1, integrating (9 we get Now = t 1 µ{x : x 2 > 1/t}dt ( t 1 (1 f(t y dt dγ(y. (1 t 1 µ{x : x 2 > 1/t} dt t 1+ µ{x : x 2 > t} dt. and, by Lemma 2, the limit of the left-hand side of (1 as is equal to µ( \ {}. On the other hand, the functions y t 1 (1 f(t y dt (11 are uniformly bounded by 2. Write these functions as We have ( y 1 t 1 (1 f(t y dt = 1 y y / y / = 1 ( y 1 t 1 f(tdt y / t 1 f(tdt t 1 f(tdt. y / y / t 1 f(tdt t 1 f(tdt y 1 ( y, as, t 1 f(tdt ( y /, as,

1 ALEXANDER KOLDOBSKY and by our assumption t 1 f(tdt = ψ( 1, as. Therefore, the functions (11 converge to zero pointwise and are uniformly bounded by a constant. By the dominated convergence theorem, the limit of the right-hand side of (1 as is equal to. Sending in (1, we get that µ( \ {} =, therefore the probability measure µ is a unit atom at the origin and f is a constant function, which contradicts to the assumption of Theorem 1. End of the proof of Theorem 1: Let k be the sequence from Lemma 4. Recall that g is a non-negative function (see (5. By Lemmas 1, 3, 4, lim g( k = lim (u + v + w( k = ( 1 + c ln x ˆφ(xdx, k k where c < 1. Therefore, (ln, φ = ln x ˆφ(xdx for every even non-negative test function φ supported outside of the origin. By Proposition 1, (, embeds in L. 3. Examples The concept of embedding of a normed space in L was studied in [KKYY]. In particular, every finite dimensional subspace of L q, < q 2 embeds in L (see [KKYY, Th.6.7]. On the other hand, if (, K embeds in L, it also embeds in L q for every n < q <, and in particular, is a k-intersection body for every integer k, < k < n (see [KKYY,Th.6.3]; for the definition and properties of embeddings in L q, q < and their connections with geometry see [K3, Ch. 6]. Also, if (, K embeds in L q for every q (,, then it also embeds in L. However, there are examples of normed spaces that embed in every L q, q (, but don t embed in L (see [KKYY, Th. 6.5]. These results confirm the place of L in the scale of L q -spaces. Speaking informally, the space L is larger than every L q, q (, 2, but smaller than every L q, q <. The following result from [K3, Th. 4.19] provides many examples of normed spaces that do not embed in L. Here we present the conclusion of this theorem in a different but equivalent form, using Theorem 6.15 from [K3]. Note that it is essential that the space X is a normed space,

POSITIVE DEFINITE FUNCTIONS 11 so that its unit ball is a convex body. The proof, though quite involved technically, is based on the idea that the second derivative at zero of a positive definite function cannot be equal to. Because of that, we call this result the second derivative test. Proposition 2. Let n 4, n < p < and let X = (, be an n-dimensional normed space with a normalized basis e 1,..., e n so that: (i For every fixed (x 2,...,x n 1 \ {}, the function n x 1 x 1 e 1 + x i e i has a continuous second derivative everywhere on R, and i=2 x x 1 (, x 2,...,x n = x x 2 1(, x 2,...,x n =, where x x 1 and x stand for the first and second partial derivatives x 2 1 by x 1 of the norm x 1 e 1 + + x n e n. (ii There exists a constant C so that, for every x 1 R and every (x 2,...,x n 1 with x 2 e 2 + + x n e n = 1, one has (iii Convergence in the limit x x 2 1(x 1, x 2,...,x n C. lim x 1 x x 2 1(x 1, x 2,...,x n = is uniform with respect to (x 2,...,x n 1 satisfying the condition x 2 e 2 + + x n e n = 1. Then the space ( embeds in L p, p < if and only if p [n 3, n. Since every space that embeds in L also embeds in every L p, p < and any space satisfying the conditions of Theorem 2 does not embed in L p, 1 < p <, we conclude that such spaces do not embed in L. By Theorem 1, Corollary 2. If a normed space (,, n 4 satisfies the conditions of Theorem 2, then a function of the form f( can be positive definite only if f is a constant function. The norm of such a space cannot appear as the standard of an n-dimensional version. Let us mention several examples of spaces satisfying the conditions of Theorem 2. For normed spaces X and Y and q R, q 1, the q-sum (X Y q of X and Y is defined as the space of pairs {(x, y : x X, y Y } with the norm (x, y = ( x q X + y q Y 1/q.

12 ALEXANDER KOLDOBSKY It was proved in [K2, Th 2] that such spaces with q > 2 satisfy the conditions of Theorem 2 provided that the dimension of X is greater or equal to 3. Another example is that of Orlicz spaces. Recall that an Orlicz function M is a non-decreasing convex function on [, such that M( = and M(t > for every t >. The norm M of the n-dimensional Orlicz space l n M is defined implicitly by the equality n k=1 M( x k / x M = 1, x \ {}. As shown in [K2, Th 3], the spaces l n M, n 4 satisfy the conditions of Theorem 2 if the Orlicz function M C 2 ([, is such that M ( = M ( =. Note that every three dimensional normed space embeds in L (see [KKYY, Corollary 4.3], so it is impossible to extend the result of Corollary 2 to the case n = 3 using the method of this paper. However, as mentioned in the Introduction, the classes Φ(l 3 q are trivial if q > 2. References [AMM] I. Aharoni, B. Maurey and B. Mityagin, Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces, Israel J. Math. 52 (1985, 251 265. [BDK] J. Bretagnolle, D. Dacunha-Castelle and J. L. Krivine, Lois stables et espaces L p, Ann. Inst. H. Poincaré Probab. Statist. 2 (1966, 231 259. [CKS] S. Cambanis, R. Keener, and G. Simons, On α-symmetric multivariate distributions, J. Multivariate Analysis 13 (1983, 213 233. [D] L. Dor, Potentials and isometric embeddings in L 1, Israel J. Math. 24 (1976, 26 268. [E] M. Eaton, On the projections of isotropic distributions, Ann. Stat. 9 (1981, 391 4. [F] T. S. Ferguson, A representation of the symmetric bivariate Cauchy distributions, Ann. Math. Stat. 33 (1962, 1256 1266. [G] T. Gneiting, On α-symmetric multivariate characteristic functions, J. Multivariate Anal. 64 (1998, 131 147. [H] C. Herz, A class of negative definite functions, Proc. Amer. Math. Soc. 14 (1963, 67 676. [KKYY] N. J. Kalton, A. Koldobsky, V. Yaskin and M. Yaskina, The geometry of L, Canad. J. Math. 59 (27, 129 149. [K1] A. Koldobsky, The Schoenberg problem on positive-definite functions, Algebra i Analiz 3 (1991, 78 85; translation in St. Petersburg Math. J. 3 (1992, 563 57. [K2] A. Koldobsky, Second derivative test for intersection bodies, Advances in Math. 136 (1998, 15 25. [K3] A. Koldobsky, Fourier analysis in convex geometry, Amer. Math. Soc., Providence RI, 25. [K4] A. Koldobsky, A note on positive definite norm dependent functions, Proceedings of the Conference on High Dimensional Probability, Luminy, 28, to appear

POSITIVE DEFINITE FUNCTIONS 13 [KL] A. Koldobsky and Y. Lonke, A short proof of s conjecture on positive definite functions, Bull. London Math. Soc. 31 (1999, 693 699. [Ku] Yu. G. Kuritsyn, Multidimensional versions and two problems of Schoenberg, Problems of Stability of Stochastic Models, VNIISI, Moscow, 1989, 72 79. [Le] P. Lévy, Théorie de l addition de variable aléatoires, Gauthier-Villars, Paris, 1937. [L] J. Lindenstrauss, On the extension of operators with finite dimensional range, Illinois J. Math. 8 (1964, 488 499. [Li1] A. Lisitsky, One more proof of Schoenberg s conjecture, unpublished manuscript, 1991. [Li2] A. Lisitsky, The Eaton problem and multiplicative properties of multivariate distributions, Theor. Probab. Appl. 42 (1997, 618 632. [M1] J. Misiewicz, On norm dependent positive definite functions, Bull. Acad. Sci. Georgian SSR 13 (1988, 253 256. [M2] J. Misiewicz, Positive definite functions on l, Stat. Probab. Let. 8 (1989, 255 26. [M3] J. Misiewicz, Substable and pseudo-isotropic processes connections with the geometry of subspaces of L α -spaces, Dissertationes Math. (Rozprawy Mat. 358 (1996. [R] D. St. P. Richards, Positive definite symmetric functions on finite dimensional spaces. 1.Applications of the Radon transform, J. Multivariate Analysis 19 (1986, 28 298. [S1] I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938, 522 536. [S2] I. J. Schoenberg, Metric spaces and completely monotone functions, Annals of Math. 39 (1938, 811 841. [VTC] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, Probability distributions on Banach spaces, D. Reidel Publishing Company, Dordrecht, 1987. [Z1] V. Zastavnyi, Positive definite norm dependent functions, Dokl. Russian Acad. Nauk. 325 (1992, 91 93. [Z2] V. Zastavnyi, Positive definite functions depending on the norm, Russian J. Math. Phys. 1 (1993, 511 522. Department of Mathematics, University of Missouri, Columbia, MO 65211 E-mail address: koldobsk@math.missouri.edu