Positivity and convergence in fermionic quantum field theory

We derive norm bounds that imply the convergence of perturbation theory in fermionic quantum field theory if the propagator is summable and has a fini te Gram constant. These bounds are sufficient for an application in renorma lization group studies. Our proof is conceptually simple and technically el ementary; it clarifies how the applicability of Gram bounds with uniform co nstants is related to positivity properties of matrices associated to the p rocedure of taking connected parts of Gaussian convolutions. This positivit y is preserved in the decouplings that also preserve stability in the case of two-body interactions.