  This paper considers a massive random access problem in which a large number
of sporadically active devices wish to communicate with a base station (BS)
equipped with massive multiple-input multiple-output (MIMO) antennas. Each
device is preassigned a unique signature sequence, and the BS identifies the
active devices by detecting which sequences are transmitted. This device
activity detection problem can be formulated as a maximum likelihood estimation
(MLE) problem for which the sample covariance matrix of the received signal is
a sufficient statistic. The goal of this paper is to characterize the feasible
set of problem parameters under which this covariance based approach is able to
successfully recover the device activities in the massive MIMO regime. Through
an analysis of the asymptotic behaviors of MLE via its associated Fisher
information matrix, this paper derives a necessary and sufficient condition on
the Fisher information matrix to ensure a vanishing probability of detection
error as the number of antennas goes to infinity, based on which a numerical
phase transition analysis is obtained. This condition is also examined from a
perspective of covariance matching, which relates the phase transition analysis
to a recently derived scaling law. Further, we provide a characterization of
the distribution of the estimation error in MLE, based on which the error
probabilities in device activity detection can be accurately predicted.
Finally, this paper studies a random access scheme with joint device activity
and data detection and analyzes its performance in a similar way.
