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A plane transverse electromagnetic wave propagating antiparallel to a cold relativistic electron beam extending over all space is backscattered from a negative energy space‐charge wave into a frequency up‐converted plane transverse electromagnetic wave. Parametric and nonlinear three‐wave interactions are analytically examined in a systematic manner taking account of all the nonlinearities. With the inclusion of appropriate phenomenological collision terms, and frequency and wave‐number mismatches, the amplitude threshold, the growth rate, and the saturation values are determined. A heuristic estimation reveals wave amplification to be limited by the beam velocity. An expression for maximum amplification is obtained. Representative numerical results are presented to display the characteristics of the nonlinear three‐wave interaction.