The beam-plasma interaction which has been proposed by Stix is examined both by approximate analysis and by integration of the non-linear equations of motion which describe the interaction of an individual electron with a monochromatic large amplitude electrostatic plasma wave. The quasi-stochastic model of the large amplitude plasma waves, introduced by Stix (1964), has been used in the calculations by programming a random phase function into the argument of the periodic plasma wave function. It is found that electrons are subject to a quasi- cyclotron acceleration which proceeds to a higher energy in the stochastic case than the limit found in the non-stochastic case (a phase-coherent plasma wave). This behaviour is interpreted in terms of a limit cycle phenomenon, and the stochastic phase shift appears to push electrons across these limit cycles. It is also found that favoured groups of electrons in a 2eV plasma excited by a 5 keV beam in a2000 Gauss field can achieve energies in the range 85 to 170 keY in less than a nanosecond by a single acceleration occurring in one plasma wave coherence length L‖ as defined by Stix.