Bursty waves with widely varying electric fields persist over a wide range of altitudes ~2--7 RE at polar cap latitudes in Earth's inner magnetosphere. These PF waves have frequencies near expected values of the electron plasma frequency (PF), well below the electron gyrofrequency, and are believed to be generated by electron beams. Here it is demonstrated that stochastic growth theory (SGT) can account well for the detailed field statistics of the archetypal PF wave event, by showing that the observed distributions P(log E) of envelope electric fields E are well fitted by the lognormal function predicted by SGT. Weak evidence exists that a nonlinear wave process coexists with stochastic growth physics at large fields ≳1 mV m-1 on the basis of fits of the observed P(log E) distribution to the SGT prediction that includes a nonlinear process at high fields and on the discovery of a class of low-frequency waves that may be produced by nonlinear decay of PF waves. An analytic model is developed for why the waves evolve to an SGT state, on the basis of waves driven by an electron beam in MHD density turbulence and the development of fluctuations in the electron beam due to wave growth occurring in localized regions. The model is viable for the polar cap plasma parameters considered, predicting that the wave burstiness should be of order that observed and that the beam fluctuations should have timescales ~10 ms that are below current detection capabilities. SGT thus accounts well for the burstiness and wave statistics, as well as the persistence of the waves and driving distribution. The consistency of the PF wave statistics with SGT also implies the presence of local electron beams over much of the polar cap, whose source is currently unknown. This application brings to five the number of contexts in which SGT applies, suggesting that SGT is widely applicable in space plasmas. ¿ 2001 American Geophysical Union