Expansion of the Scale Mixture of the Multivariate Normal Distribution with Error Bound Evaluated in the L1-Norm

Let Z be a random vector following the p-variate normal distribution N(0, Ip), and let S be a positive definite random matrix independent of Z. The probability density function f(x) of the random vector X = S1/2Z is expanded around that of N(0, Ip) and its error bound is evaluated in terms of the L1-norm. The bound is given in the form Ck,pE tr(S − I)k, where Ck,p is a constant depending only on k, the number of terms of the expansion, and p.