  We analyze general enough models of repeated indirect measurements in which a
quantum system interacts repeatedly with randomly chosen probes on which Von
Neumann direct measurements are performed. We prove, under suitable hypotheses,
that the system state probability distribution converges after a large number
of repeated indirect measurements, in a way compatible with quantum wave
function collapse. Similarly a modified version of the system density matrix
converges. We show that the convergence is exponential with a rate given by
some relevant mean relative entropies. We also prove that, under appropriate
rescaling of the system and probe interactions, the state probability
distribution and the system density matrix are solutions of stochastic
differential equations modeling continuous-time quantum measurements. We
analyze the large time convergence of these continuous-time processes and prove
convergence.
