Simple analytical models have proved very useful in understanding vibration energy harvesters driven by a sinusoidal acceleration. Corresponding analyses for broadband excitations have been absent. In this paper, we present new closed-form results on the output power, proof mass displacement, and optimal load of linear resonant energy harvesters driven by broadband vibrations. Output power dependence on signal bandwidth is also considered. The results are compared with those that are already well established for a sinusoidal acceleration. We formulate a stochastic description of more general energy-harvester models and show that the influence of elastic mechanical stoppers on the output power is dependent on the electrical load for large amplitude vibrations.