  This paper studies the asymptotic properties of average area spectral
efficiency (ASE) of a downlink cellular network in the limit of very dense base
station (BS) and user densities. This asymptotic analysis relies on three
assumptions: (1) interference is treated as noise; (2) the BS locations are
drawn from a Poisson point process; (3) the path loss function is bounded above
satisfying mild regularity conditions. We consider three possible definitions
of the average ASE, all of which give units of bits per second per unit
bandwidth per unit area. When there is no constraint on the minimum operational
signal-to-interference-plus-noise ratio (SINR) and instantaneous full channel
state information (CSI) is available at the transmitter, the average ASE is
proven to saturate to a constant, which we derive in a closed form. For the
other two ASE definitions, wherein either a minimum SINR is enforced or CSI is
not available, the average ASE is instead shown to collapse to zero at high BS
density. We provide several familiar case studies for the class of considered
path loss models, and demonstrate that our results cover most previous models
and results on ultradense networks as special cases.
