The order of a univariate elliptical distribution (Q1199004)

Let \(Y\) be a random variable (r.v.) with density \(h(y^ 2)\), \(E(Y)=0\), \(E(Y^ 2) =1\), and let \(K(x)= 2^{-1} \int_{x^ 2}^ \infty h(v)dv\). A r.v. \(X= \mu+ \sigma Y\) \((\sigma>0)\) is said to have an elliptical density (e.d.) with parameters \(\mu\) and \(\sigma\) and its \(M\)- function is defined by \(M(x)= \sigma^ 2 K(t)/ h(t^ 2)\), with \(t= (x- \mu)/ \sigma\). The author shows that \[ M(x)\geq 0, \quad M(\mu+ x)= M(\mu- x) \quad \text{and} \quad E[ M(X)]= \sigma^ 2. \] Two characterizations of an e.d. in terms of the \(M\)-functions are given. A r.v. \(\sigma Y\) is said to have an e.d. of order \(p\) if its \(M\)-function is a polynomial of order \(p\). It is shown that: (i) an e.d. of odd order does not exist, (ii) an e.d. of order zero corresponds to the normal distribution and (iii) an e.d. of order two corresponds to a generalized Student \(t\) distribution. An explicit expression for an e.d. of order four is also derived.