A note on Gaussian distributions in R n

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1 Proc. Indian Acad. Sci. (Math. Sci. Vol., No. 4, November 0, pp c Indian Academy of Sciences A note on Gaussian distributions in R n B G MANJUNATH and K R PARTHASARATHY Indian Statistical Institute, Delhi Centre, 7, S. J. S. Sansanwal Marg, New Delhi 0 06, India bgmanjunath@gmail.com; krp@isid.ac.in MS received August 0; revised 9 October 0 Abstract. Given any finite set F of (n -dimensional subspaces of R n we give examples of nongaussian probability measures in R n whose marginal distribution in each subspace from F is Gaussian. However, if F is an infinite family of such (n - dimensional subspaces then such a nongaussian probability measure in R n does not exist. Keywords. Gaussian distribution; characteristic function; homogeneous polynomial; linear functionals; nonunimodality; Hermite polynomial.. Introduction Starting with the simple example of E Nelson as cited by Feller in [] wehavefrom[ 4], etc., as well as 0 of [5], several examples of bivariate and multivariate nongaussian distributions under which many linear functionals can have a Gaussian distribution on the real line. These results suggest the possibility of characterizing a Gaussian distribution in R n through properties of classes of linear functionals. Motivated by Nelson s example in [] and the bivariate construction in [] we introduce a perturbation of the standard Gaussian density function in R n exhibiting the following interesting features: ( Given any finite set {S j, j N} of (n -dimensional subspaces it has a marginal density function which is the standard Gaussian in each S j, j {,,...,N}; ( There can exist linear functionals whose distributions may have nonunimodal density functions; (3 For certain choices of subspaces the nongaussian perturbation can be so chosen that any real symmetric measurable function of all the n co-ordinates has its distribution preserved. In particular, the sum of squares of all the co-ordinates can have the χ distribution with n degrees of freedom. We also demonstrate the following characterization of the multivariate Gaussian distribution. Suppose {S j, j =,,...} is a countably infinite set of (n -dimensional subspaces of R n and μ is a probability measure in R n such that the projection of μ in each subspace S j is Gaussian. Then μ itself is Gaussian. This is a generalization of the characterization in [] and a more precise version of the result in [4]. 635

2 636 B G Manjunath and K R Parthasarathy Our proofs follow the steps in [] and we use some additional geometric and topological arguments of a very elementary kind.. A perturbation of the Gaussian characteristic function We begin by examining a small perturbation of the characteristic function of the n-variate standard Gaussian distribution with mean vector 0 and covariance matrix I as follows. Choose and fix any homogeneous polynomial P of even degree k in n real variables t, t,...,t n and define (t; ε,, P = e t + ε e t P(t, t R n (. where t = (t,...,t n T, ε is a real parameter and is a parameter satisfying 0 < <. Here t = (t + +t n. Clearly, ( ; ε,, P is a real analytic function on R n satisfying Let (0; ε,, P =, ( t; ε,, P = (t; ε,, P. (. Z P ={t P(t = 0, t R n } (.3 be the set of zeros of P in R n. Since we are interested in the inverse Fourier transform of we introduce the renormalized polynomial : P : in the form of a formal definition. DEFINITION. Let N(x = π e x and let H m (x be the m-th Hermite polynomial defined by d m dx m N(x = ( m H m (xn(x, m = 0,,,... (as in []. For any real polynomial P in n real variables given by P(t, t,...,t n = m a m,m,...,m n t m tm...t m n n its renormalized version : P : is defined by : P : (x,...,x n = m a m,m,...,m n H m (x H m (x...h mn (x n. Note that for a homogeneous polynomial, its renormalized version need not be homogeneous.

3 A note on Gaussian distributions in R n 637 Since the function in (.isinl (R n its inverse Fourier transform f is defined by First, we note that f (x; ε,, P = = (π n ( π n e + ε (π n e itt x (t; ε,, Pdt dt...dt n x e itt x e t P(tdt...dt n. (.4 (π n e itt x e t dt dt...dt n = n n ( x j N. j= Repeated differentiation with respect to x, x,...,x n shows that for the homogeneous polynomial P of degree k we have (π n e itt x e t P(tdt dt...dt n = ( n P n ( x j i,...,i N x x n = ( k ( n+k : P : x,..., x n j= ( π n e x. Thus the inverse Fourier transform (.4 assumes the form f (x; ε,, P = ( π n e x { + ( k ε n+k : P : ( x,..., x } n e x (. (.5 Since, by assumption, > 0 the positive constant K (, P defined by K (, P = sup x R n :P : (x,...,x n n+k e x ( (.6 is finite and for all x R n, f (x; ε,, P 0, if ε K (, P. We observe that ( ; ε,, P is a real characteristic function of the probability density function f ( ; ε,, P defined by (.5 for any ε [ K (, P, K (, P]. Herewe have made use of property (.. Thus we can summarize the discussion above as a theorem.

4 638 B G Manjunath and K R Parthasarathy Theorem.. Let 0 < <, P be a real homogeneous polynomial in n variables of even degree k, K (, P the positive constant defined by (.6 and ε [ K (, P, K (, P]. Then the function ( ; ε,, P defined by (. is the characteristic function of a probability density function f ( ; ε,,p defined by (.5. Under this density function f ( ; ε,, P the linear functional x a T x with a = has characteristic function ϕ a and probability density function g a on the real line given respectively by ϕ a (t = e t + ε P(ae t t k, t R (.7 g a (x = { e x + ( k ε P(a π k+ H k ( x e x}. (.8 In particular, for any a Z P, the linear functional a T x has the normal distribution with mean 0 and variance a but f ( ; ε,, P is a nongaussian density function for any ε [ K (, P, K (, P]\{0}. Proof. The first part is immediate from the discussion preceding the statement of the theorem. To prove the second part, we note that the characteristic function ϕ a (t of the linear functional a T x under the density function f ( ; ε,, P is (ta; ε,, P and (.7 follows from (. and the homogeneity of P. Now(.8 follows from Fourier inversion of (.7. If 0 = a Z P, then 0 = P(a = P ( a a and therefore ϕ a a (t = e t. Hence a T x is normally distributed with mean 0 and variance a. COROLLARY.3 Let {S j, j N} be any finite set of (n -dimensional subspaces of R n. Then there exists a nongaussian analytic probability density function whose projection on S j is Gaussian for each j {,,..., N}. Proof. By adding one more (n -dimensional subspace to the collection {S j, j N}, if necessary, we may assume without loss of generality that N is even. Choose a unit vector a ( j S j for each j and define the homogeneous real polynomial P of degree N by P(t = N a ( jt t, t R n. j= Clearly, P(t = 0, if t N S j. j=

5 A note on Gaussian distributions in R n 639 In other words, N S j Z P. j= If we choose μ to be the probability measure with the density function f ( ; ε,, P,0 = ε in [ K (, P, K (, P] in Theorem. it follows immediately from the last part of the theorem that every linear functional of the form b T x has a normal distribution with mean 0 and variance b whenever b Z P. This completes the proof. Remark.4. In the context of understanding the modes of the density function g a (x given by (.8, it is of interest to note that {x x = 0, g a (x = 0} { = x x = 0, e x ( + ( k εp(a H k+ ( x k+ x } = 0. Indeed, this is obtained by straightforward differentiation and using the recurrence relation H k+ (x = xh k (x H k (x. Example.5. Let n be even and P(t, t,...,t n = t t...t n (ti t j = t t...t n i> j... t t... tn t 4 t 4... tn 4... tn (n t (n t (n. (.9 Then P is a polynomial of even degree n, which is antisymmetric in the variables t, t,...,t n. The renormalized version : P : of P is given by H (x H (x... H (x n H 3 (x H 3 (x... H 3 (x n : P : (x, x,...,x n =. (.0 H n+ (x H n+ (x... H n+ (x n In particular, : P : is antisymmetric in the variables x, x,...,x n. Fixing 0 < < we get for each ε [ K (, P, K (, P], with K (, P being determined

6 640 B G Manjunath and K R Parthasarathy by (.6, (.0 and k = n, the probability density function f ( ; ε,, P is given by f (x; ε,, P = { ( e π n x + ε n(n+ ( H x ( H x 3 H ( x... H ( xn ( H x 3... H 3 ( xn H n+ ( x Hn+ ( x... Hn+ ( xn e x }. (. By Theorem. and its corollary we conclude that the projection of this density function on the (n -dimensional hyperplanes {x x j = 0}, j n; {x x i x j = 0}, i j n and {x x i + x j = 0}, i j n are all (n -dimensional Gaussian densities. If g(x, x,...,x n is any bounded continuous function which is symmetric in the variables x, x,...,x n then the function g : P : is an antisymmetric function in R n and therefore (g : P :(x, x,...,x n e x dx dx...dx n = 0. R n Thus g(x,...,x n f (x; ε,, Pdx dx...dx n R n = g(x, x,...,x n R n ( π n e x dx dx...dx n. In other words, for 0 = ε [ K (, P, K (, P], any symmetric measurable function g on R n has the property that its distribution under the nongaussian density function f (x; ε,, P in (. is the same as its distribution under the standard Gaussian density function with mean 0 and covariance matrix I. Example.6. We now specialize Example.5 to the case n =, = / when P(t, t = t t (t t, : P : (x, x = H 3 (x H (x H (x H 3 (x = x 3 x x 3 x. A simple computation shows that K (, P = 8sup x 3 x x 3 x e 4 (x +x = 8e.

7 A note on Gaussian distributions in R n 64 This supremum is easily evaluated by switching over to the polar co-ordinates x = r cos θ, x = r sin θ. Then f (x; ε,, P = π e (x +x{ + 3ε(x 3 x x 3 x e (x +x} (. which is a probability density function whenever ε e 8. At ε = 0, it is the standard normal density function with mean 0 and covariance matrix I. We write η = 3ε and express the density function (. as f η (x, x = π e (x +x{ + η(x 3 x x 3 x e (x +x} (.3 where (see figure. η e 4. When a = (sin θ,cos θ the density function g θ of the linear functional x x sin θ + x cos θ, under f η is given by the formula (.8 of Theorem. as g θ (x = } e η sin(4θ { x (4x 4 x + 3e x. (.4 π 3 Thus g θ (x = x } e {e x η sin(4θ x (4x 4 0x + 5e x. π f x x Figure. Bivariate density f η (x, x at η = e /4.

8 64 B G Manjunath and K R Parthasarathy θ=0.57 o θ=6.87 o g θ g θ x x θ=4.3 o θ=54.43 o g θ g θ Figure. Nonunimodality of g θ. x x It is not difficult to find values of η in the range (0, 4 e ] and angle θ for which { x } e η sin(4θ x (4x 4 0x + 5 = 0, x = 0 =. 6 (.5 This reveals the possibility of nonunimodality of the density of some linear functionals under the joint density f η. For an illustration, see figure. 3. A characterization of Gaussian distributions in R n In the context of the Corollary to Theorem. we have the following characterization of a Gaussian distribution in R n when the number N of (n -dimensional subspaces in the corollary is countably infinite. Theorem 3.. Let {S j, j =,,...} be a countably infinite set of (n -dimensional subspaces of R n and let μ be a probability measure in R n whose projection on S j is Gaussian for each j =,,... Then μ is Gaussian. Proof. The fact that the projection of μ on the two distinct (n -dimensional subspaces S and S are Gaussian implies that the multivariate Laplace transform ˆμ of μ given by μ(z,...,z n = exp(z x + +z n x n μ(dx dx...dx n (3.

9 A note on Gaussian distributions in R n 643 is well-defined for z C n and analytic in each of the complex variables z j, j =,...,n. Let m and be respectively the mean vector and covariance matrix of the R n valued random variable x with distribution μ. Choose and fix a unit vector a ( j S j for each j =,,... Suppose Since a ( jt = (a j,...,a jn, j =,,..., α j = max a jr. r n n a jr =, r= j it follows that α j n /, j. There exists an r 0 such that a jr0 = α j for infinitely many values of j. Restricting ourselves to this infinite set of j s and assuming r 0 = without loss of generality we may as well assume that a ( j = (a j,...,a jn T, a j = max a jr, j =,,..., r n a j n /, j. Now consider the (n -dimensional vector b ( j defined by ( a b ( jt j =, a j3,..., a jn, j =,,... a j a j a j where a jr, r =, 3,...,n. a j Thus all the vectors b ( j are distinct and they constitute a bounded countable set in R (n. Define the set D = { s } s R (n,(b ( j b (i T s = 0. j<i Being a countable intersection of dense open sets it follows from the Baire category theorem that D is dense in R (n. Let now s = (s, s 3,...,s n T R (n be any fixed point in D. Define s j = b ( jt s, j =,,... By the definition of D, {s j, j =,,...} is a bounded and countably infinite set of distinct points on the real line. Furthermore, a j s j + a j s + +a jn s n = 0, j.

10 644 B G Manjunath and K R Parthasarathy In other words, (s j, s,...,s n T S j for each j. By hypothesis the linear functional s j x + s x + +s n x n has a normal distribution with mean s j m + s m + +s n m n and variance (s j, s,...,s n (s j, s,...,s n T. Defining ( ψ(z,...,z n = exp m T z + zt z, z C n we conclude that the Laplace transform ˆμ defined by (3. and the function ψ satisfy the relation ˆμ(s j, s,...,s n = ψ(s j, s,...,s n for j =,,... Since ˆμ(z, s,...,s n and ψ(z, s,...,s n are analytic functions of z in the whole complex plane and they agree on the infinite bounded set {s j, j =,,...} it follows that ˆμ(z, s,...,s n = ψ(z, s,...,s n, z C. Since this holds for all (s,...,s n T D which is dense in R (n and both sides of the equation are continuous on R n we have ˆμ(s, s,...,s n = ψ(s, s,...,s n for all (s, s,...,s n T R n. This implies that μ is a Gaussian measure with mean vector m and covariance matrix. Acknowledgements We thank S Ramasubramanian for bringing our attention to the example of E. Nelson on page 99 of [] and J Stoyanov for useful suggestions. References [] Feller W, An Introduction to Probability Theory and Its Applications (000 (India: Wiley vol. II, nd edition [] Hamedani G G and Tata M N, On the determination of the bivariate normal distribution from distributions of linear combinations of the variables, The Amer. Math. Mon. 8 ( [3] Kale B K, Normality of linear combinations of non-normal random variables, The Amer. Math. Mon. 77 ( [4] Shao Y and Zhou M, A characterization of multivariate normality through univariate projections, J. Multi. Anal. 0 ( [5] Stoyanov J, Counterexamples in Probability (997 (New York: Wiley nd edition