On some properties of elliptical distributions Fredrik Armerin∗ Abstract We look at a characterization of elliptical distributions in the case when finiteness of moments of the random vector is not assumed. Some addi- tional results regarding elliptical distributions are also presented. Keywords: Elliptical distributions, multivariate distributions. JEL Classification: C10. ∗CEFIN, KTH, Sweden. Email: [email protected] 1 Introduction Let the n-dimensional random vector X have finite second moments and the property that the distribution of every random variable on the form hT X +a for every h 2 Rn and a 2 R is determined by its mean and variance. Chamberlain [2] showed that if the covariance matrix of X is positive definite, then this is equivalent to the fact that X is elliptically distributed. There are, however, elliptical distributions that do not have even finite first moments. In this note we show that for a random vector to be elliptically distributed is equivalent to it fulfilling a condition generalizing the moment condition above, and one that can be defined even if the random vector do not have finite moments of any order. In portfolio analysis, if r is an n-dimensional random vector of returns and there is a risk-free rate rf , then the expected utility of having the portfolio n (w; w0) 2 R × R is given by T E u w r + w0rf ; where u is a utility function (we assume that this expected value is well defined). See e.g. Back [1], Cochrane [3] or Munk [5] for the underlying theory. If the T distribution of w r + rf only depends on its mean and variance, then T T T E u w r + w0rf = U w µ + w0rf ; w Σw (1) for some function U (this is one of the applications considered in Chamberlain [2]). If we only consider bounded utility functions, then the expected value is well defined even if r does not have any finite moments. Below we will show that if Equation (1) holds for any bounded and measurable function u, then this is in fact a defining property of elliptical distributions, i.e. r must be elliptically distributed if Equation (1) holds for any bounded and measurable u. The results presented in this note are previously known, or are basic gener- alizations of known results. 2 Basic definitions The general reference for this section is McNeil et al [4]. Definition 2.1 An n-dimensional random vector X has a spherical distribution if UX =d X for every orthogonal n × n matrix U, i.e. for every n × n matrix U such that UU T = U T U = I: The following is Theorem 3.19 in McNeil [4]. 1 Theorem 2.2 Let X be an n-dimensional random vector. The following are equivalent. (i) The random vector X has a spherical distribution. (ii) There exists a function of a scalar variable such that h T i E eih X = (hT h) for every h 2 Rn. (iii) For every h 2 Rn T d h X = khkX1; where khk2 = hT h. We call in the theorem above the characteristic generator of X, and write X ∼ Sn( ) if X is n-dimensional and has a spherical distribution with characteristic generator . For a strictly positive integer n we let (ek) for k = 1; : : : ; n denote the vectors n of the standard basis in R . If X ∼ Sn( ) for some characteristic generator , then by choosing first h = ek and then h = −ek we get from property (iii) in Theorem 2.2 that d −Xk = Xk; i.e. each component in a random vector which has a spherical distribution is a symmetric random variable. By choosing first h = ek and then h = e` and again using (iii) in Theorem 2.2, we get d Xk = X` for every k; ` = 1; : : : ; n, i.e. every component of a spherically distributed ran- dom vector has the same distribution. Definition 2.3 An n-dimensional random vector X is said to be elliptically distributed if X =d µ + AY; n T where µ 2 R , A is an n × k-matrix and Y ∼ Sk( ). With Σ = AA we write X ∼ En(µ, Σ; ) in this case. The characteristic function of X ∼ En(µ, Σ; ) is given by h T i T E eih X = eih µ (hT Σh): If X has finite mean, then µ = E [X] ; and if X has finite variance, then we can choose Σ = Var(X): 2 If X ∼ En(µ, Σ; ), B is an k × n-matrix and b is an k × 1-dimensional vector, then T BX + b ∼ Ek(Bµ + b; BΣB ; ): Alternatively, if X ∼ En(µ, Σ; ), then BX + b =d Bµ + BAY; T where Y ∼ Sn( ) and AA = Σ. Finally, when Σ is a positive definite matrix we have the equivalence −1=2 X ∼ En(µ, Σ; ) , Σ (X − µ) ∼ Sn( ): 3 Characterizing elliptical distributions The following proposition shows the structure of any elliptically distributed random vector. Proposition 3.1 Let µ 2 Rn and let Σ be an n × n symmetric and positive semidefinite matrix. For an n-dimensional random vector X the following are equivalent. (i) X ∼ En(µ, Σ; ). (ii) We have p hT X =d hT µ + hT Σh Z for any h 2 Rn, where Z is a symmetric random variable with E eitZ = (t2): n Proof. (i) ) (ii): If X ∼ En(µ, Σ; ), then for every h 2 R and some matrix A such that AAT = Σ hT X =d hT µ + hT AY = hT µ + (AT h)T Y =d hT µ + kAT hkY p 1 = hT µ + hT AAT hY p 1 T T = h µ + h ΣhY1: Since Y has a spherical distribution, Y1 is a symmetric random variable with characteristic function E eitY1 = (t2): (ii) ) (i): If X has the property that p hT X =d hT µ + hT Σh Z 3 n itZ 2 for every h 2 R and where E e = (t ), then p h T i T h T i T E eih X = eih µE ei h Σh Z = eih µ (hT Σh); i.e. X ∼ En(µ, Σ; ). 2 Note that the previous proposition is true even if Σ is only a positive semidefinite matrix. Remark 3.2 With the same notation as in Proposition 3.1, if the random vec- tor X has the property that for every h 2 Rn p hT X =d hT µ + hT Σh Z holds and Σ has at least one non-zero diagonal element (the only case when this does not hold is when Σ = 0), then Z must be symmetric. To see this we assume, without loss of generality, that Σ11 > 0. Now first choose h = e1, and then h = −e1. We get d p d p X1 = µ1 + Σ11 Z and − X1 = −µ1 + Σ11 Z respectively, or X − µ X − µ p1 1 =d Z and − p1 1 =d Z Σ11 Σ11 respectively. It follows that Z =d −Z. Using the representation p hT X =d hT µ + hT Σh Z we see that the finiteness of moments of the vector X is equivalent to the finiteness of the moments of the random variable Z. This representation is also a practical way of both defining new and understanding well known elliptical distributions. When Z ∼ N(0; 1) we get the multivariate normal distribution, and when Z ∼ t(ν) we get the multivariate t-distribution with ν > 0 degrees of freedom. The multivariate t-distribution with ν 2 (0; 1] is an example of an elliptical distribution which does not have finite mean. Now assume that the random vector X has the property that the distribution of hT X + a is determined by its mean and variance for every h 2 Rn and a 2 R. If we let µ = E [X] and Σ = Var(X), which we assume is a positive definite matrix, then Chamberlain [2] showed that X must be elliptically distributed. Hence in this case, if T T T T E h1 X + a1 = E h2 X + a2 and Var(h1 X + a1) = Var(h2 X + a2); then we must have T d T h1 X + a1 = h2 X + a2: 4 This property can, with notation as above, be rewritten as follows: If T T T T h1 µ + a1 = h2 µ + a2 and h1 Σh1 = h2 Σh2; then T d T h1 X + a1 = h2 X + a2: It turns out that this condition, which is well defined for any X ∼ En(µ, Σ; ) even if no moments exists, is a defining property of elliptical distributions if Σ is a positive definite matrix. Proposition 3.3 Let µ 2 Rn and let Σ be an n × n symmetric and positive definite matrix. For an n-dimensional random vector X the following are equiv- alent. (i) X ∼ En(µ, Σ; ) for some characteristic generator . (ii) For any measurable and bounded f : R ! R and any h 2 Rn and a 2 R E f(hT X + a) = F (hT µ + a; hT Σh) for some function F : R × R+ ! R. (iii) If T T T T h1 µ + a1 = h2 µ + a2 and h1 Σh1 = h2 Σh2 n for h1; h2 2 R and a1; a2 2 R, then T d T h1 X + a1 = h2 X + a2: For a proof of this, see Section A.1. It is possible to reformulate this proposition without using the constants a, a1 and a2. Proposition 3.4 Let µ 2 Rn and let Σ be an n × n symmetric and positive definite matrix.