Approximations of density functions are considered in the multivariate
case. The results are presented with the help of matrix derivatives,
powers of Kronecker products and Taylor expansions of functions with m
atrix argument. In particular, an approximation by the Wishart distrib
ution is discussed. It is shown that in many situations the distributi
ons should be centred. The results are applied to the approximation of
the distribution of the sample covariance matrix and to the distribut
ion of the non-central Wishart distribution.