Asymptotic convergence of the partial averaging technique

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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/n²) if the potential has first order Sobolev derivatives. If the potential has second order Sobolev derivatives, the convergence is shown to be O(1/n³) and we give the exact expressions for the convergence constants of the density matrices.