  The cumulant representation of the Fourier path integral method is examined
to determine the asymptotic convergence characteristics of the imaginary-time
density matrix with respect to the number of path variables $N$ included. It is
proved that when the cumulant expansion is truncated at order $p$, the
asymptotic convergence rate of the density matrix behaves like $N^{-(2p+1)}$.
The complex algebra associated with the proof is simplified by introducing a
diagrammatic representation of the contributing terms along with an associated
linked-cluster theorem. The cumulant terms at each order are expanded in a
series such that the the asymptotic convergence rate is maintained without the
need to calculate the full cumulant at order $p$. Using this truncated
expansion of each cumulant at order $p$, the numerical cost in developing
Fourier path integral expressions having convergence order $N^{-(2p+1)}$ is
shown to be approximately linear in the number of required potential energy
evaluations making the method promising for actual numerical implementation.
