On the PDF of the sum of random vectors

There are various cases in physics and engineering sciences (especially com munications) where one requires the envelope probability density function ( pdf) of the sum of several random sinusoidal signals. According to the corr espondence between a random sinusoidal signal and a random vector, the sum of random vectors can be considered as an abstract mathematical model for t he above sum. Now it is desired to obtain the pdf of the length of the resu lting vector. Considering the common and reasonable assumption of uniform d istributions for the angles of vectors, many:researchers have obtained the pdf of the length of the resulting vector only for special cases. However i n this paper, the pdf is obtained for the most general case in which the le ngths of vectors are arbitrary dependent random variables. This pdf is in t he form of a definite integral, which may be inappropriate for analytic man ipulations and numerical computations. So an appropriate infinite Laguerre expansion is also derived. Finally the results are applied to solve a typic al example in computing the scattering cross section of random scatterers.