  We performed systematic studies on the effects of event-by-event efficiency
fluctuations on efficiency correction for cumulant analysis in relativistic
heavy-ion collision experiments. Experimentally, particle efficiencies of
events measured under different experimental conditions should be different.
For fluctuation measurements, the final event-by-event multiplicity
distributions should be the superposed distributions of various type of events
measured under different conditions. We demonstrate efficiency fluctuation
effects using numerical simulation, in which we construct an event ensemble
consisting of events with two different efficiencies. By using the mean
particle efficiencies, we find that the efficiency corrected cumulants show
large deviations from the original inputs when the discrepancy between the two
efficiencies is large. We further studied the effects of efficiency
fluctuations for the cumulants of net-proton distributions by implementing the
UrQMD events of Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV in a realistic STAR
detector acceptance. We consider the unequal efficiency in two sides of the
Time Projection Chamber (TPC), multiplicity dependent efficiency, and the
event-by-event variations of the collision vertex position along the
longitudinal direction ($V_\mathrm{z}$). When the efficiencies fluctuate
dramatically within the studied event sample, the effects of efficiency
fluctuations have significant impacts on the efficiency corrections of
cumulants with the mean efficiency. We find that this effect can be effectively
suppressed by binning the entire event ensemble into various sub-event samples,
in which the efficiency variations are relatively small. The final efficiency
corrected cumulants can be calculated from the weighted average of the
corrected factorial moments of the sub-event samples with the mean efficiency.
