On the conditional variance for scale mixtures of normal distributions (Q1587358)

Suppose \({\mathbb X}\) is a scale mixture of normal vectors, that is \({\mathbb X}=A^{1/2}{\mathbb G}\), where \({\mathbb G\in R^n}\) is a Gaussian random vector, \(n>1\), and \(A\) is a positive variable independent of \({\mathbb G}\). Write \({\mathbb X}=({\mathbb X_1,\mathbb X_2})\) where \({\mathbb X_1\in R^m}\) and \(1\leq m<n\). The authors establish that the conditional second moment of \({\mathbb X_2}\) given \({\mathbb X_1}\) is always finite a.s. for \(m\geq 2\), and it is a.s. finite for \(m=1\) if and only if \(E[A^{1/2}]<\infty\). An expression is derived for the conditional covariance Cov\(({\mathbb X_2 |\mathbb X_1})\) in terms of the distribution of \({\mathbb X}\) and the Laplace transform of the distribution of \(A\). Applications to uniform and stable scales of normal distributions are also given.