Multivariate quadratic forms of random vectors

We obtain the distribution of the sum of n random vectors and the distribution of their quadratic forms: their densities are expanded in series of Hermite and Laguerre polynomials. We do not suppose that these vectors are independent. In particular, we apply these results to multivariate quadratic forms of Gaussian vectors. We obtain also their densities expanded in Mac Laurin series or in the form of an integral. By this last result, we introduce a new method of computation which can be much simpler than the previously known techniques. In particular, we introduce a new method in the very classical univariate case. We remark that we do not assume the independence of normal variables.