On the conditional variance for scale mixtures of normal distributions

Pot a scale mixture of normal vector, X = A(1/2)G, where X, G is an element of R-n and A is a positive variable, independent of the normal vector G, w e obtain that the conditional valiance covariance, Cov(X-2 \ X-1), is alway s finite a.s. for m greater than or equal to 2, where X-1 is an element of R-n and m < n, and remains a.s. finite even form = 1, if and only if the sq uare root moment of the scale factor is finite. It is shown that the varian ce is not degenerate as in the Gaussian case, but depends upon a function S -A,S-m (.) for which various properties are derived. Application to a unifo rm and stable: scale of normal distributions are also given. (C) 2000 Acade mic Press. AMS 1991 subject classifications: 60E07, 60E10, 62B10, 62J05.