  The effect on parametric instability growth of pump wave incoherence is
treated by deriving a set of equations governing the space-time evolution of
the ensemble-average coupled-mode amplitudes and intensities. Particular
attention is paid to establishing the regions of validity of the statistical
description. Thresholds, growth rates, and amplification rates are given for
both spatially and temporally incoherent pump waves. Both absolutely and
convectively unstable modes are considered. The statistical results are
verified where appropriate by numerical integration of the coupled-mode
equations with different models of pump incoherence.
