We analyze a class of vector multiple access channels with additive noise, where the sum of the dimensions of the transmitted signals matches that of the received signal. We first focus on the case without power constraints, using point process techniques. We derive the capacity region in the Poltyrev sense, a representation of the error probabilities for each subset of transmitters based on Palm theory, and random coding exponents for each type of error event in the case without power constraints, focusing on the case of independent and identically distributed Gaussian noise, with arbitrary positive definite covariance matrix at each time. This also leads to random coding error exponents in the traditional power-constrained case, where the power constraint at each transmitter is defined by an arbitrary positive definite matrix at each time.