On the conditional variance for scale mixtures of normal distributions

For a scale mixture of normal vector, X = A 1/2 G , where X ,  G ∈ R n and A is a positive variable, independent of the normal vector G , we obtain that the conditional variance covariance, Cov( X 2 ∣ X 1 ), is always finite a.s. for m ⩾2, where X 1 ∈ R n and m < n , and remains a.s. finite even for m =1, if and only if the square root moment of the scale factor is finite. It is shown that the variance is not degenerate as in the Gaussian case, but depends upon a function S A ,  m (·) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also given. Keywords heteroscedasticity, stable random vectors, marginal densities