  The counting statistics of electron transport is theoretically studied in a
system with two capacitively coupled parallel transport channels. Each channel
is composed of a quantum dot connected by tunneling to two reservoirs. The
nonequilibrium steady state of the system is controlled by two affinities or
thermodynamic forces, each one determined by the two reservoirs of each
channel. The status of a single-current fluctuation theorem is investigated
starting from the fundamental two-current fluctuation theorem, which is a
consequence of microreversibility. We show that the single-current fluctuation
theorem holds in the limit of a large Coulomb repulsion between the two
parallel quantum dots, as well as in the limit of a large current ratio between
the parallel channels. In this latter limit, the symmetry relation of the
single-current fluctuation theorem is satisfied with respect to an effective
affinity that is much lower than the affinity determined by the reservoirs.
This back-action effect is quantitatively characterized.
