On the Conditional Variance for Scale Mixtures of Normal Distributions

For a scale mixture of normal vector, X=A1/2G, where X, G∈Rn and A is a positive variable, independent of the normal vector G, we obtain that the conditional variance covariance, Cov(X2 ∣ X1), is always finite a.s. for m⩾2, where X1∈Rn 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 SA, m(·) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also given.