The Hubbard-Stratonovich transformation allows one to formulate the problem of calculating the ground state properties of a many-body theory as one of sampling a distribution. This distribution is constructed by propagating a trial wave function under the influence of a one-body time-dependent external field. However, a straightforward application of the Hubbard-Stratonovich transformation gives distributions which are not always positive definite for a generic trial wave function. In this work it is rigorously shown that, for the Hubbard model, in many cases the non-positiveness of this distribution is not important for reaching the infinite imaginary time limit, i.e., the ground state properties.