  I define and characterize the reweighted methods, which are techniques used
in conjunction with the random series implementation of the Feynman-Kac
formula. I prove several convergence results valid for all series
representations and then I specialize the results for the Levy-Ciesielski and
Wiener-Fourier series. As opposed to the partial averaging method on which they
are based, the reweighted techniques do not involve any modification of the
physical potential. Rather, the underlying idea is to develop some specialized
constructions of the Brownian bridge that enters the Feynman-Kac formula, so as
to simulate the partial averaging effect. For the Levy-Ciesielski series
representation, I develop a reweighted technique which has o(1/n^2) convergence
for potentials having first order Sobolev derivatives. It is suggested that the
asymptotic convergence may reach O(1/n^3) for potentials having second order
Sobolev derivatives. The method preserves the favorable log(n) scaling for the
time necessary to compute a path at a given discretization point. For the
Wiener-Fourier series representation, the particular reweighted method designed
in the present article is shown to have O(1/n^3) convergence if the potential
has second order Sobolev derivatives. The convergence constant has superior
dependence with the inverse temperature as compared to the partial averaging
method for the same series. Because the expression of the convergence constant
does not actually involve the second order derivatives of the potential, it is
conjectured that the O(1/n^3) convergence extends to the potentials having
first order Sobolev derivatives only.
