We discuss conditions under which expectation values computed from a c
omplex Langevin process Z will converge to integral averages over a gi
ven complex-valued weight function. The difficulties in proving a gene
ral result are pointed out. For complex-valued polynomial actions, it
is shown that for a process converging to a strongly stationary proces
s one gets the correct answer for averages of polynomials if c(tau)(k)
=E(e(ikZ(tau))) satisfies certain conditions. If these conditions are
not satisfied, then the stochastic process is not necessarily describe
d by a complex Fokker-Planck equation. The result is illustrated with
the exactly solvable complex frequency harmonic oscillator.