We report the polarity reversal of a magnetic vortex core using a nonresonant in-plane pulsed magnetic field of arbitrary waveform studied using time-resolved x-ray photoemission electron microscopy and micromagnetic simulations. The imaging and simulations show that a 5 mT pulse, higher than the critical field for nonlinear effects, effectively leads to the randomization of the vortex core polarity. The micromagnetic simulations further show that the onset of stochastic core polarity randomization does not necessarily coincide with the critical reversal field, leading to a field window for predictable core reversal.