Error bounds for asymptotic expansion of the conditional variance of the scale mixtures of the multivariate normal distribution

Let X = A(1/2) G be a scale mixture of a multivariate normal distribution with X, G is an element of R-n, G is a multivariate normal vector, and A is a positive random variable independent of the multivariate random vector G. This study presents asymptotic results of the conditional variance-covariance, Cov(X-2 \ X-1), X-1 is an element of R-m, m < n, under some moment expressions. A new representation form is also presented for conditional expectation of the scale variable on the random vector X-1 is an element of R-m, m < n. Both the asymptotic expression and the representation are manageable and in computable form. Finally, an example is presented to illustrate how the computations are carried out.