  We study the asymptotic convergence of the partial averaging method, a
technique used in conjunction with the random series implementation of the
Feynman-Kac formula. We prove asymptotic bounds valid for most series
representations in the case when the potential has first order Sobolev
derivatives. If the potential has also second order Sobolev derivatives, we
prove a sharper theorem which gives the exact asymptotic behavior of the
density matrices. The results are then specialized for the Wiener-Fourier
series representation. It is found that the asymptotic behavior is o(1/n^2) if
the potential has first order Sobolev derivatives. If the potential has second
order Sobolev derivatives, the convergence is shown to be O(1/n^3) and we give
the exact expressions for the convergence constants of the density matrices.
