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The history-dependent recurrence theory for multiplication noise in avalanche photodiodes (APDs), developed by Hayat et al., is generalized to include inter-layer boundary effects in heterostructure APDs with multilayer multiplication regions. These boundary effects include the initial energy of injected carriers as well as bandgap-transition effects within a multilayer multiplication region. It is shown that the excess noise factor can be significantly reduced if the avalanche process is initiated with an energetic carrier, in which case the initial energy serves to reduce the initial dead space associated with the injected carrier. An excess noise factor reduction up to 40% below the traditional thin-APD limit is predicted for GaAs, depending on the operational gain and the multiplication-region's width. The generalized model also thoroughly characterizes the behavior of dead space as a function of position across layers. This simultaneously captures the effect of the nonuniform electric field as well as the anticipatory nature of inter-layer bandgap-boundary effects.