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We consider the stochastic process stemming from continuously measuring a
quantum system with multiple jump channels. The process is described not only
by the random times between jumps, but also by a sequence of emitted symbols
representing each jump channel. We establish the fundamental properties of this
sequence of symbols. First, we determine a special set of superoperators that
completely govern the dynamics, and provide an efficient way for computing
multi-point distributions and for simulating stochastic trajectories. We also
determine the conditions for the stochastic process to be stationary and show
that the memory between distant emissions is determined by the spectral
properties of a specific superoperator. Finally, we show that some systems
support a pattern, where the evolution after each jump runs over a closed set
of states. This, as we argue, can be used to greatly facilitate our prediction
of future outcomes. We illustrate these ideas by studying transport through a
boundary-driven one-dimensional XY spin chain. We show that the statistics
depends dramatically on the chain size. And that the presence of pairing terms
in the Hamiltonian destroy any existing patterns.
