Fluctuations in time of the intensity of indirect exciton photoluminescence in GaAs/Al0.33Ga0.67As double quantum wells have been studied. An analysis of the behavior of the intensity fluctuations under variation in external control parameters, such as the temperature and the voltage applied perpendicular to the structure, has revealed that they directly reflect the manifestation of Bose condensation. It has been demonstrated that the current flowing perpendicular to the structure also undergoes pronounced fluctuations simultaneously with those of the intensity. A study of possible reasons that could account for the onset of such fluctuations suggests the need for taking into account tunneling transitions in the exciton Bose condensate system.